Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 670 https://internationalpubls.com Relation Between the Fractional Domination Number of Some Graphs and Their Line Graphs Mahesh Sarada1,2 , Rekha Jain1 , Ganesh Mundhe3 1Department of Mathematics, Medi-Caps University, Pigdamber, Rau-453331, Indore, Madhya Pradesh, India. 2Department of Engineering Mathematics, Pimpri Chinchwad College of Engineering & Research, Ravet-412101, Pune, Maharashtra, India. 3Department of Engineering Mathematics, Army Institute of Technology, Dighi-411015, Pune, Maharashtra, India. Article History: Received: 27-05-2024 Revised: 20-07-2024 Accepted: 30-07-2024 Abstract: Let G is the simple connected graph with order n and its line graph noted as L(G). Let the fractional dominating number denoted by γ_(f ) (G) of graph G and the upper fractional dominating number Г_(f ) (G). In this study we have obtained union and join on〖 γ〗_(f ) (G) and γ_(f ) (L(G)) exploring the graphs for upper fractional domination number including Cycle, complete, Star, Bi-Star graph, wheel graph, Cubic graph, Graphs of Cartesian product like (K_2 ×〖 P〗_(n )), (K_3 ×〖 P〗_(n )) and (Cm×Cn) with consideration of the computational complexity. We have taken parameters related to fractional domination in line graphs towards generalization. The goal of this paper is to provide a generalized results of sum 〖 γ〗_(f ) (G)+γ_(f ) (L(G)), Г_(f ) (G)+Г_(f ) L(G) and product 〖 γ〗_(f ) (G)*γ_(f ) (L(G)), Г_(f ) (G)*Г_(f ) L(G) for some specific graph classes. Keywords: Domination set, fractional dominating number, line graph, union and join. 2020 Mathematical Sciences Classification: 05C72, 05C76, 05C90. 1. Introduction Initially, the idea of graph domination was presented by author Ore in the year 1960, and after this it is very important and vital area of research in theory of graph. The research regarding fractional domination number presented by some researchers in the late 1980s as a generalized concept of domination. A graph's fractional dominating number is described as minimum total weights assigned to all the vertices of graph from fractional dominating set, where a fractional dominating set is a method that gives each vertex a weight such that the addition of the weights of all vertices is at least one. At the 18th south-eastern international conference based on Combinatorics, Graph theory and Computing S.T. Hedetniemi initially defined fractional dominating as a function 𝑓:𝑉 → [0, 1] is said to be dominating function of the graph 𝐺 = (𝑉, 𝐸) if 𝑓(𝑁[𝑣]) ≥ 1 for all 𝑣 ∈ 𝑉. The fractional dominating number of 𝐺 is given by 𝛾𝑓 (𝐺). M. Farber examined the issue of figuring out whether to use the linear programming formulation of 𝛾𝑓 (𝐺) could provide an integer solution to LPP. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 671 https://internationalpubls.com 𝛾𝑓 (𝐺) = 𝑀𝐼𝑁 ∑ 𝑥𝑖 , 𝑛 𝑖=1 Subject to 𝑁.𝑋 ≥ 1𝑛 , where 𝑥𝑖 > 0. The constraint system for domination is 𝑁.𝑋 ≥ 1𝑛 and using a closed neighborhood matrix N we aim to reduce the weight of the function that accomplishes this with 𝑛-by-𝑛 binary matrix 𝑁 = 𝐴 + 𝐼𝑛 , where 𝐼𝑛 be the 𝑛-by-𝑛 identity matrix and 𝐴 is adjacency matrix. The authors [12] have discuss topological spaces generated by simple graphs using adjacency relation and non adjaceny relation on vertices. They establish important results showing relations between complete graph and discrete topological space. Also discuss the topological spaces related to complete graphs, isomorphic graphs and study their properties. The author [8] have taken the simple connected graph 𝐺 of 𝑛 vertices and 𝑚 edges. Let 𝐿(𝐺) is its line graph, In this paper they have obtained lower bound and upper bound for addition of graph and its line graph’s domination number. They continue to work on the block graph and further domination number of graph. In the paper [9] they characterize regular graphs and unicyclic graphs of odd order for which 𝛾 (𝐺) + 𝛾(𝐿 (𝐺)) = 𝑛 − 2 . The authors [1] have studied another fractional version of k- domination in the graphs and related parameters. Let 𝐺 is the connected graph and k is the positive number with 𝑘 ≤ 𝑟𝑎𝑑(𝐺). A subset 𝐷 ⊆ 𝑉 is known as a distance k-domination set of graph 𝐺 if f every 𝑣 ∈ 𝑉 − 𝐷, there exists a vertex 𝑢 ∈ 𝐷 such that 𝑑 (𝑢, 𝑣) ≤ 𝑘. The line graph of a graph represents the adjacencies between the line connections of the original graph. This study investigates relationship between the fractional domination number of a graph and their line graph for various graph classes. Our results show that there is a correlation between the sum of fractional domination number of a graph and its line graph, and this sum is NP-hard in general, however the correlation between the sum of the fractional domination number of a graph 𝐺 and its line graph 𝐿(𝐺) is not straightforward. Generally, depends on the specific graphs involved or some graph classes, it can be solved in polynomial time. These findings contribute to our understanding of the properties of graphs. In some graph classes, the fractional domination number of 𝐿(𝐺) can be related to the fractional domination number of 𝐺. If 𝐺 has a high fractional domination number, this might imply that 𝐿(𝐺) also has a relatively high fractional domination number due to the structure and relationship between the two graphs. Positive Correlation: In some cases, the fractional domination number of 𝐿(𝐺) can be related to the fractional domination number of 𝐺. If 𝐺 has a high fractional domination number, this might imply that 𝐿(𝐺) also has a relatively high fractional domination number due to the structure and relationship between the two graphs. No Simple Correlation: However, there is no universal rule stating that the sum of the fractional domination numbers of 𝐺 and 𝐿(𝐺) must be correlated in a simple manner. The relationship is influenced by various factors such as the size of 𝐺, its connectivity, and how vertices in 𝐿(𝐺) are dominated by sets of vertices in 𝐺. The goal of this paper is to investigate a generalization of 𝛾𝑓 (𝐺), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 672 https://internationalpubls.com 𝛾𝑓 (𝐿(𝐺)) and Г𝑓 (𝐺), Г𝑓 𝐿(𝐺) for some specific graphs. In this section we are going to give definitions which are useful in further sections. The definitions are given [5, 7] like the following: Definition 1.1. Domination set and domination number: Let 𝐺 = (𝑉, 𝐸) is a graph and subset 𝐷 of vertex set 𝑉 is known as a dominating set of graph if every vertex in 𝑉 − 𝐷 is adjacent to at least one vertex in subset 𝐷. A dominating set 𝐷 is known as a minimal dominating set if there isn't a dominating set of 𝐺 that is a proper subset of D. The size of smallest dominating set of graph 𝐺 is called the domination number of graph 𝐺 and it is denoted by 𝛾 (𝐺). The maximum number of elements of minimal dominating set of graph 𝐺 is known as upper domination number of graph and it is denoted by Г(G) The domination number noted as 𝛾 (𝐺) and upper domination number noted as Г(G) are described as: γ(G) = 𝑚𝑖𝑛 {|𝐷|: 𝐷 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑠𝑒𝑡 𝑜ƒ 𝐺}, and Г(G) = 𝑚𝑎𝑥 {|𝐷|: 𝐷 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑠𝑒𝑡 𝑜ƒ 𝐺}. Definition 1.2. Fractional dominating function: A dominating function 𝑓 to be any 𝑓: 𝑉(𝐺) → [0, 1] is a function of graph 𝐺 which allocate values for every vertex 𝑣 ∈ 𝑉(𝐺) in unit interval [0,1]. The function 𝑓 is known as fractional dominating function if for every vertex v ∈ 𝑉(𝐺), 𝑓(𝑁[𝑣]) = ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑣] ≥ 1. It means the addition of the values allocated to the vertices in closed neighborhood of 𝑣 ∈ 𝑉(𝐺) such that 𝑁[𝑣] is at least one, i.e. (𝑁[𝑣]) ≥ 1. (Since then any vertex 𝑣 ∈ 𝑉(𝐺) is in the closed neighborhood of at least one vertex in 𝐷, where 𝐷 is subset of vertex set 𝑉). Definition 1.3. Fractional domination number: The dominating function 𝑓 is known as a minimal fractional domination function if there doesn’t exist another domination function 𝑔 ≠ 𝑓 for which 𝑔(𝑣) ≤ 𝑓(𝑣) for all 𝑣 ∈ 𝑉(𝐺) equivalently 𝑓 as an minimal fractional dominating function if for each vertex 𝑣 with 𝑓(𝑣) > 0, there exist a vertex 𝑤 ∈ 𝑁[𝑣] such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1 By lemma [8] Let 𝑓 be a dominating function for a graph 𝐺 = (𝑉, 𝐸) then 𝑓 is minimal dominating if and only if for any 𝑣 ∈ 𝑉(𝐺) whenever 𝑓 (𝑣) > 0 there exists some 𝑢 ∈ 𝑁[𝑣] such that 𝑓(𝑁[𝑣]) = 1. (If there is a vertex 𝑣 for which given condition is not true, means if every vertex in the closed neighborhood of 𝑣 carry out (𝑁[𝑣]) > 1, then we are in a position to decrease 𝑓 (𝑣) to obtain a smaller fractional dominating function and so 𝑓 is not a minimal fractional dominating function. The fractional domination number of 𝐺, 𝛾𝑓 (𝐺) and upper fractional domination number of 𝐺, Г𝑓 (𝐺) are described as, 𝛾𝑓 (𝐺) = 𝑚𝑖𝑛 {|𝑓|: 𝑓 𝑖𝑠 𝑎𝑛 𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐺}, Г𝑓 (𝐺) = 𝑚𝑎𝑥 {|𝑓|: 𝑓 𝑖𝑠 𝑎𝑛 𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐺}. where |𝑓| = ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑉 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 673 https://internationalpubls.com Definition 1.4. Line graph: Line graph of 𝐺 is 𝐿 (𝐺) having its vertices are converted as the edges of original graph 𝐺 with two vertices of graph 𝐿 (𝐺) are adjacent for the condition if corresponding edges share a common vertex in graph 𝐺. Another definition that the vertices of 𝐿 (𝐺) are the edges of graph 𝐺 with two vertices are adjacent whenever the corresponding edges of 𝐺 are adjacent. In any minimal dominating set 𝑆 is subset of 𝑉 (𝐺), such that 𝑁[𝑆] = 𝑉(𝐺) and the closed neighborhood of each 𝑣 ∈ 𝑆 contains at least one vertex and which is not in the closed neighborhood of any other member of 𝑆 . (If this is not true for some 𝑣 ∈ 𝑆 then 𝑆 − {𝑣} is a smaller dominating set, contradicting the minimality of 𝑆). Consequently, every maximal irredundant set is likewise a minimal dominant set. Hence 𝑖𝑟 (𝐺) ≤ 𝛾(𝐺) earlier this result was obtained by [3] every minimal dominating set induces a minimum dominating function, and every minimal dominating function is said to be a minimal fractional dominating function. This implies that 𝛾𝑓 (𝐺) ≤ 𝛾(𝐺) ≤ Г(G) ≤ Г𝑓 (𝐺). Some preliminary known results: Theorem 1.1. ([5]). For every graph 𝐺 we have ⌈ 𝑛 1+∆(𝐺) ⌉ ≤ 𝛾 (𝐺) ≤ 𝑛 − 𝛥(𝐺). Theorem 1.2. ([5]). For every graph 𝐺 we have 𝑛 1+∆(𝐺) ≤ 𝛾𝑓 (𝐺) ≤ 𝑛 1+𝛿(𝐺) , Where ∆(𝐺) 𝑖𝑠 noted as maximum degree and 𝛿(𝐺) is noted as minimum degree of graph G. Theorem 1.3. ([5]). For any graph 𝐺 fractional domination number 𝛾𝑓 (𝐺) = 1 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝛥(𝐺) = 𝑛 − 1. Corollary 1.1. ([5]). Let 𝐺 is any simple connected graph with 𝛥(𝐺) = 𝑛 − 1 and let 𝑓 be any fractional domination function with (𝐺) = |𝑓|. Then for only those vertices 𝑣 ∈ 𝑉(𝐺) with 𝑑𝑒𝑔 (𝑣) = 𝑛 − 1 satisfy 𝑓(𝐺) > 0. Theorem 1.4. ([3]). If the graph 𝐺 is r-regular then 𝛾𝑓 (𝐺) = 𝑛 𝑟+1 . In section 2 we determine the fractional and upper fractional dominating number of graph 𝐺 and 𝐿(𝐺). We also determine the 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) , 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) and Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)), Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)). We have taken the standard graphs like Cycle, Complete, Star, Bi- Star graph, Wheel graph, Cubic graph, Cartesian product graphs. In section 3 we introduce applications in terms of union and join of the considered graphs. 2. Relation between the fractional domination number of graph and their line graph 2.1 Cycle graph (𝐶𝑛) Simple graph 𝐺 of order 𝑛 where (𝑛 ≥ 3) and number of edges 𝑛 is known as a cycle graph. Every edge in the graph creates a cycle with length 𝑛 and each vertex has degree two. The line graph of cycle graph is 𝐿(𝐺) isomorphic to its cycle graph 𝐺 with same vertex numbers and are adjacent in the provided graph if the corresponding edges share a common vertex. Theorem 2.1.1. If 𝐺 is the simple connected cycle graph (𝐶𝑛) on 𝑛 vertices and 𝐿(𝐺) is line graph, then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 674 https://internationalpubls.com i) 2 ≤ 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) ≤ 2𝑛 3 ii) 1 ≤ 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) ≤ 𝑛 2/9 iii) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 𝑛 iv) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 𝑛 2/4 Proof: i) and ii) with the reference of previous chapter for cycle graph we have fractional domination number is 𝛾𝑓 (𝐶𝑛 ) = 𝑛/3, 𝑤ℎ𝑒𝑟𝑒 𝑛 = 3,4,5, … 𝑛. The line graph of cycle graph is 𝐿(𝐺) isomorphic to its cycle graph 𝐺 with same vertex numbers and are adjacent in the provided graph if the corresponding edges share a common vertex. Therefore, for line graph 𝐿(𝐺) the fractional dominating number is 𝛾𝑓 (𝐿(𝐶𝑛 )) = 𝑛/3. For the sum 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) lower bound is 2 and upper bound is 2𝑛 3 . Therefore we have 2 ≤ 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) ≤ 2𝑛 3 and 1 ≤ 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) ≤ 𝑛2/9, 𝑛 = 3,4,5, … 𝑛. iii) and iv) The upper fractional domination number of cycle graph 𝐶𝑛 is Г𝑓 (𝐶𝑛 ) = 𝑛/2 is observed by assigning weight (½) to every vertex and the condition where vertex 𝑤 ∈ 𝑁[𝑣] such that∑ 𝑓(𝑣)⬚ 𝑣∈𝑁[𝑤] = 1 condition is satisfied. Therefore we found that Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 𝑛 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 𝑛 2/4. □ Theorem 2.1.2 Let graph 𝐺 is 𝑟-regular of order 𝑛 then its upper fractional domination number is Г𝑓 (𝐺) ≤ 𝑛/𝑟 Proof: Theorem 1.4. gives if graph 𝐺 be 𝑟-regular of order 𝑛 then 𝛾𝑓 (𝐺) = 𝑛 𝑟+1 . The fractional dominating function 𝑓 that allocates the weight 1/(𝑟 + 1) to each vertex of 𝐺 so that function is the minimal weight fractional dominating function, and its generalized form is 𝛾𝑓 (𝐺) = 𝑛 𝑟+1 . So An identical argument works for given upper fractional domination number where function 𝑓 allocating weights (1/𝑟) to each vertex of 𝐺 so that Г𝑓 (𝐺) is maximum cardinality of function 𝑓 where 𝑓 is minimal fractional dominating function of 𝐺. Its generalized form is given by Г𝑓 (𝐺) ≤ 𝑛/𝑟. Let 𝑣 ∈ 𝑉(𝐺) such that 𝑓(𝑁[𝑉]) = ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑣] ≥ 1. Г𝑓 (𝐺) = 𝑚𝑎𝑥 {|𝑓|: 𝑓 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑎𝑙 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐺}. Where |𝑓| =∑ 𝑓(𝑣) ⬚ 𝑣∈𝑉 . So if graph 𝐺 is 𝑟-regular of order 𝑛 then its upper fractional domination number is Г𝑓 (𝐺) ≤ 𝑛/𝑟 . □ 2.2 Complete graph (𝐾𝑛) A simple connected graph 𝐺 is considered to complete if every pair of vertices are adjacent. A complete graph 𝐺 is simple undirected graph in which pair of different vertices is connected by a unique edge. In the line graph of complete graph number of vertices in 𝐿(𝐺) are equal in number of edges in graph 𝐺. https://en.wikipedia.org/wiki/Simple_graph https://en.wikipedia.org/wiki/Undirected_graph https://en.wikipedia.org/wiki/Vertex_(graph_theory) https://en.wikipedia.org/wiki/Edge_(graph_theory) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 675 https://internationalpubls.com Theorem 2.2.1. Let graph 𝐺 is a complete on 𝑛 vertices and 𝐿(𝐺) is its line graph on 𝑛(𝑛−1) 2 vertices then i) 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛 − 1)/2 (2𝑛 − 4) + 1 ii) 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1) 2 (2𝑛−4)+1 iii) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛 − 1)/2 2𝑛 − 4 iv) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1)/2 2𝑛−4 Proof: i) and ii) If graph 𝐺 is the complete then all pair of other vertices are connected by a unique edge and the fractional dominating number 𝛾𝑓 (𝐾𝑛 ) = 1. Any set containing a single vertex is a dominating set in 𝐾𝑛. So 𝛾 (𝐺) = 1. For any graph 𝐺 we have 𝛾𝑓 (𝐺) ≤ 𝛾 (𝐺). The graph 𝐺 is a complete on 𝑛 vertices as 𝑛 − 1 regular graph we have Г𝑓 (𝐾𝑛) ≤ 𝑛 𝑛−1 . The line graph 𝐿(𝐺) is having its vertices are nothing but edges of 𝐺 with two vertices of 𝐿(𝐺) are adjacent if common vertex shared by corresponding edges in 𝐺. In the line graph of complete graph number of vertices in 𝐿(𝐺) are equal in number of edges in graph 𝐺. (See the Figure 2.2.1A and 2.2.1B) where 𝐾4 is complete graph with 4 vertices and 6 edges so in its line graph of 𝐾4 we have 6 vertices with each vertex of degree 4. So 𝐿(𝐾4) is 4-regular graph. In line graph of Complete Graph 𝐾4 we have every two nonadjacent vertices are mutually adjacent to exactly four vertices. (A) Complete graph 𝐾4 (B) Line graph of Complete graph 𝐾4 Figure 2.2.1 Similarly, 𝐾5 is complete with 5 vertices and 10 edges so in the line graph of 𝐾5 we get 10 vertices with each vertex of degree 6. For the generalized result we have the sequence of complete graph 𝐾𝑛 with set of vertices as {1,2,3,4,5,6,7, … 𝑛}. So for complete graph on set of vertices 𝑛 = {2,3,4,5, … , 𝑛} the sequence formed for graphs with number of edges {1,3,6,10,15,21,28, … 𝑛(𝑛−1) 2 }, is triangular number sequence. For 𝐺 is complete graph with 𝑛 vertices as 𝑛 − 1 regular connected graph where 𝑛 = {2,3,4,5,6, … }, the line graph of complete graph 𝐿(𝐺) on 𝑛(𝑛−1) 2 vertices are (2𝑛 − 4) regular connected graph. Line graph having each vertex with degree sequence is {0,2,4,6,8,10,…2𝑛 − 4}. So https://en.wikipedia.org/wiki/Vertex_(graph_theory) https://en.wikipedia.org/wiki/Edge_(graph_theory) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 676 https://internationalpubls.com we have 𝛾𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1) 2 (2𝑛−4)+1 hence for the sum 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛−1) 2 (2𝑛−4)+1 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1) 2 (2𝑛−4)+1 . iii) and iv) The condition satisfies for the complete graph on 𝑛 vertices as (𝑛 − 1) regular graph where each vertex 𝑤 ∈ 𝑁[𝑉] 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1 and it is known that Г𝑓 (𝐾𝑛) = 1. So if graph 𝐺 is complete and 𝐿(𝐺) is its line graph then upper fractional dominating number of line graph is Г𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1)/2 2𝑛−4 and for the sum we get Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛−1)/2 2𝑛−4 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1)/2 (2𝑛−4) . □ Number of Vertices Number of Edges Graph with each vertex of Degree Line Graph with each vertex of Degree 2 1 1 𝛾𝑓 (𝐺) = 1 0 𝛾𝑓 (𝐿(𝐺)) = 1 3 3 2 𝛾𝑓 (𝐺) = 1 2 𝛾𝑓 (𝐿(𝐺)) = 1 4 6 3 𝛾𝑓 (𝐺) = 1 4 𝛾𝑓 (𝐿(𝐺)) = 1.2 5 10 4 𝛾𝑓 (𝐺) = 1 6 𝛾𝑓 (𝐿(𝐺)) = 1.42 6 15 5 𝛾𝑓 (𝐺) = 1 8 𝛾𝑓 (𝐿(𝐺)) = 1.66 … … … … … … 𝑛 𝑛(𝑛 − 1) 2 𝑛 − 1 𝛾𝑓 (𝐺) = 1 (2𝑛 − 4) 𝑛(𝑛 − 1) 2 (2𝑛 − 4) + 1 Table 2.2.1 2.3 Star graph (𝑆𝑛) Star graph 𝑆𝑛 is n o t i c e d a s complete bipartite graph l i k e (𝐾1,𝑛 ) , An interior node serves as the central vertex of a tree containg 𝑛 leaves. Theorem 2.3.1 Let 𝐺 is the star graph of order 𝑛 and 𝐿(𝐺) is its line graph then i) 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 2 ii) 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 1 iii) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑛 + 𝑛 − 1 𝑛 − 2 iv) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑛2−𝑛 𝑛−2 Proof: i) and ii) The term star graph refers to be complete bipartite graph in which each vertex in the graph belongs to one set and other vertices to the other set. One central vertex is adjacent to every other vertex. So, if 𝐺 be the Star Graph (𝐾1,5 ) then 𝛾𝑓 (𝐺) = 1 see the Figure 2.3.1A). The Line graph of (𝐾1,5 ) is complete graph see the Figure 2.3.1B) and hence the fractional domination number is one i.e. 𝛾𝑓 (𝐺) = 1. Therefore, star graph 𝑆𝑛 is o b s e r v e d a s complete bipartite graph Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 677 https://internationalpubls.com l i k e (𝐾1,𝑛 ) where 𝑛 = { 3,4,5, … } and line graphs are all complete graph with 𝑛 vertices where 𝑛 = {3,4,5,6, … } so the general result for the sum 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 2 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 1. (A) Star Graph 𝐾1,5 (B) Line graph of Star Graph 𝐾1,5 Figure 2.3.1 iii) and iv) As per observations the upper fractional dominating number of Star Graph (𝐾1,5 ) is 5 with considered closed neighborhood of every vertex satisfying the condition where the vertex 𝑤 ∈ 𝑁(𝑣) such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1. By assigning weight zero to the central vertex and weight one to remaining vertices. Upper fractional dominating number of Line graph of 𝐾1,5 is one. A Star graph 𝑆𝑛 is the complete bipartite graph like (𝐾1,𝑛 ) where 𝑛 = 3,4,5, … we have complete graph on 𝑛 vertices where 𝑛 = {3,4,5,6,… } so the generalized result for 𝐺 be the star graph 𝑆𝑛 on n vertices and 𝐿(𝐺) be its line graph on 𝑛 − 1 vertices with 𝑛 − 2 regular graph then the sum Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑛 + 𝑛−1 𝑛−2 and product Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑛2−𝑛 𝑛−2 . □ 2.4 Bi-Star graph The Bi -s ta r g raph i s two copies of the star graph or two copies of (𝐾1,𝑛 ) with a tree's core vertex being an internal node and 𝑛 leaves. Theorem 2.4.1. If simple graph 𝐺 is the Bi-Star of 𝑛 vertices and graph 𝐿(𝐺) be its line graph then i) 𝛾 𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 3 ii) 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 2 iii) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 2𝑛 + 2 iv) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 4𝑛 Proof i) and ii): ([11]) A bi-star graph is created by combining the apex vertices with two copies of a star graph. The set of vertices of 𝐵𝑛,𝑛 , 𝑉 (𝐵𝑛,𝑛 ) = {𝑢, 𝑣, 𝑢𝑖, 𝑣𝑖 | 1 ≤ 𝑖 ≤ 𝑛}, where 𝑢, 𝑣 are center vertices and 𝑢𝑖, 𝑣𝑖 are all pendent vertices. The set of edges of 𝐵𝑛,𝑛 is 𝐸 (𝐵𝑛,𝑛 ) = {𝑢𝑣, 𝑢𝑢𝑖, 𝑣𝑣𝑖 | 1 ≤ 𝑖 ≤ 𝑛}. So, |𝑉 (𝐵𝑛,𝑛 )| = 2𝑛 + 2 and |𝐸 (𝐵𝑛,𝑛 )| = 2𝑛 + 1. The fractional dominating number of Bi- star graph will be 𝛾𝑓 (𝐺) = 2 by assigning weight one to central vertices and weight zero to remaining vertices (see the Figure 2.4.1A). The given line graph of Bi-star (𝐾1,3 ) is connected graph with two Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 678 https://internationalpubls.com copies of complete graph connected by one point so this graph is one point connectivity graph (see the Figure 2.4.1B). A vertex connected graph is one with a minimal number of vertices whose removal which results in a disconnected graph. We get line graph of Bi-star graph is one connected graph therefore 𝛾𝑓 (𝐿(𝐺)) = 1. In general for Bi-star graph or two copies of (𝐾1,𝑛 ) where 𝑛 = {3,4,5,6,7, … } we get 𝛾𝑓 (𝐾1,𝑛 ) = 2 and 𝛾𝑓 (𝐿(𝐾1,𝑛 )) = 1. For the sum we have 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 3 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 2. (A) Bi-Star graph 𝐾(1,3) (B) Line graph of Bi-Star graph 𝐾(1,3) Figure 2.4.1 iii) and iv) The upper fractional dominating number of (𝐾1,3 ) is Г𝑓 ((𝐾1,3 ) ) = 6 with consideration of each vertex's closed neighborhood that satisfies the requirement where the vertex 𝑤 ∈ 𝑁[𝑉] such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1 by assigning weight zero to the central vertices and weight one to remaining vertices. The upper fractional dominating number of Line graph of (𝐾1,3 ) is Г𝑓 (𝐿(𝐾1,3 ) ) = 2 by assigning weight one to vertex 𝑒1 and 𝑒5 for remaining vertices weight zero so that adjacency relation will satisfied. In general for Bi-star graph or two copies of (𝐾1,𝑛 ) where 𝑛 = {3,4,5,6,7… } we get result for 𝐺 be the Bi-star graph (𝐾1,𝑛 ) on 𝑛 vertices its Г𝑓 ((𝐾1,𝑛 ) ) = 2𝑛 and line graph of Bi-star graph (𝐾1,𝑛 ) on 𝑛 − 1 vertices its Г𝑓 (𝐿(𝐾1,𝑛 ) ) = 2 then for sum Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 2𝑛 + 2 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 4𝑛. □ 2.5 Wheel Graph (𝑊𝑛) To create a wheel graph a cycle graph 𝐶𝑛−1 ,is used, by adding one vertex or 𝑊𝑛 = 𝐶𝑛−1 + 𝐾1 . Every vertex is linked to every other vertex, forming a hub of 𝐶𝑛−1 . The line graph of wheel graph 𝐿(𝑊𝑛) contains a complete graph 𝐾𝑛−1 with cycle of 𝑛 − 1 vertices where each vertex of the cycle is connected to two vertices of 𝐾𝑛−1 . Theorem 2.5.1 If 𝐺 is wheel graph on 𝑛 vertices with size 𝑚 and 𝐿(𝐺) be its line graph of 𝑚 vertices then 𝑖) 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 1 + 𝑚 𝛿(𝐺)+1 , Where 𝛿 (𝐺) is minimum degree of 𝐿 (𝐺) and 𝑖𝑖) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 2 + 𝑛−1 2 Proof i): [11] A wheel graph of 𝑛 vertices contains a cycle graph of length 𝑛 − 1 and all the vertices of the cycle are connected to a single central vertex. Thus, wheel graph on 𝑛 vertices denoted by 𝑊𝑛 for 𝑛 > 3 is acquired by connecting a vertex to every vertex of cycle graph. The fractional dominating Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 679 https://internationalpubls.com number of wheel graph will be 𝛾𝑓 (𝑊𝑛) = 1. Wheel graph of 𝑛 vertices contains 𝑚 edges so the line graph of wheel graph 𝐿 (𝑊𝑛) is a graph in which vertices corresponds to the edges of the original wheel graph so it contains 𝑚 vertices with a complete graph 𝐾𝑛−1 and the cycle of 𝑛 − 1 vertices where every vertex of the cycle is adjacent to other two vertices of 𝐾𝑛−1. The max degree of line graph 𝛥 (𝐺) = 𝑛 and min degree of line graph 𝛿 (𝐺) = 4. For example i f wheel graph 𝐺 = 𝑊4 with number of vertices 𝑛 = 4 and number of edges 𝑚 = 6 then line graph 𝐿(𝐺) contains 𝑚 = 6 vertices with a complete graph 𝐾4-1 and the cycle of 𝑛 − 1 = 3 vertices where every vertex of the cycle is adjacent to two vertices of 𝐾4-1. We know that 𝛾𝑓 (𝑊4) = 1 and the line graph 𝐿(𝐺) where 𝐺 = 𝑊4 is of order 𝑚 = 6 and 𝛿(𝐺) = 4 we get 𝛾𝑓 (𝐿(𝑊4)) = 𝑚 𝛿(𝐺)+1 ≤ 6/5 by assigning weight (1/5) for each vertex of line graph. If wheel graph 𝐺 = 𝑊5 is of order 𝑛 = 5 and size 𝑚 = 8 then line graph 𝐿(𝐺) contains 𝑚 = 8 vertices with a complete graph 𝐾5-1 and cycle of 𝑛 − 1 = 4 vertices where each vertex of cycle is connected to two vertices of 𝐾5-1. We found that if 𝛾𝑓 (𝑊5) = 1 and its line graph of order 𝑚 = 8 and 𝛿(𝐺) = 4 then 𝛾𝑓 (𝐿(𝑊5)) = 𝑚 𝛿(𝐺)+1 ≤ 8/5 by assigning weight (1/5) for each vertex of line graph. If wheel graph 𝐺 = 𝑊6 is of order 𝑛 = 6 and size 𝑚 = 10 (See the Figure 2.5.1A) then line graph 𝐿(𝐺) contains 𝑚 = 10 vertices with complete graph 𝐾6-1 and the cycle of 𝑛 − 1 = 5 vertices where each vertex of cycle is connected to two vertices of 𝐾6-1 (See the Figure 2.5.1B). We found if 𝛾𝑓 (𝑊6) = 1 its line graph of order 𝑚 = 10 and 𝛿(𝐺) = 4 then we get 𝛾𝑓 (𝐿(𝑊6)) ≤ 𝑚 𝛿(𝐺)+1 = 2 Hence, we get the generalized result for the sum 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) ≤ 1 + 𝑚 𝛿(𝐺)+1 . (A) Wheel graph W6 (B) Line graph of Wheel graph W6 Figure 2.5.1 ii) Upper fractional dominating number of wheel Graph 𝑊𝑛 is Г𝑓 (𝑊𝑛) = 2 with consideration of the closed neighborhood of each vertex satisfying the condition where the vertex 𝑤 ∈ 𝑁[𝑉] such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1. For upper fractional dominating number of line graph of wheel Graph 𝑊𝑛 is Г𝑓(𝐿(𝑊𝑛)) ≤ 𝑛−1 2 as we have 𝑊𝑛 = 𝐶𝑛−1 + 𝐾1 and the line graph of wheel graph 𝐿(𝑊𝑛) contains a complete graph 𝐾𝑛−1 with the cycle of 𝑛 − 1 vertices where each vertex of the cycle is adjacent to two vertices of 𝐾𝑛−1 and each vertex in cycle having degree four. If we assign the weights (½) for each vertex of complete graph 𝐾𝑛−1 and each vertex in cycle having weight zero will get the result Г𝑓 (𝐿(𝑊𝑛)) ≤ 𝑛−1 2 and hence result for the sum Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 2 + 𝑛−1 2 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 680 https://internationalpubls.com 2.6 Cubic graph The regular graph with degree three is known as cubic graph. Trivalent graphs are another name for cubic graphs. The Cubic graphs with 𝑛 vertices exist only for even 𝑛 whereas the condition for a graph is to be cubic is given by 𝑚/𝑛 = 3/2, with m as size of graph and 𝑛 be the order of graph. Theorem 2.6.1 Let graph 𝐺 is the connected cubic graph or trivalent graph with 𝑛 vertices and 𝑚 edges and 𝐿(𝐺) be its line graph with order 𝑚 then i) 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 5𝑛+4𝑚 20 ii) 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 20 iii) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 4𝑛+3𝑚 12 iv) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 12 Proof i) and ii): A graph with degree three at every vertex is called a cubic graph or 3-regular graph. Cubic graph are also called trivalent graph. Cubic graphs on 𝑛 vertices exist only for even 𝑛. For example if 𝐺 be the cubic graph with 𝑛 = 6 vertices or 3-regular graph with size 𝑚 = 9 then 𝛾𝑓 (𝐺) = 𝑛 3+1 = 6/4 by the definition of fractional domination number (see the Figure 2.6.1A). Its line graph 𝐿(𝐺) has 𝑚 = 9 vertices as 4-regular graph so 𝛾𝑓 (𝐿(𝐺)) = 𝑚 4+1 = 9/5 (see the Figure 2.6.1B). The cubic graph with 𝑛 = 4 called tetrahedral graph, The cubic graph with 𝑛 = 8 called cubical graph, for all such connected cubic graphs which are 3-regular graphs of order 𝑛 = {4,6,8, … } we get the generalized result as 𝛾𝑓 (𝐺) = 𝑛 𝑟+1 where 𝑟 = 3. For its line graph 𝐿(𝐺) of order 𝑚 with (𝑟 + 1) regular graph where 𝑟 = 3, we have 𝛾𝑓 (𝐿(𝐺)) = 𝑚 𝑟+2 so for the sum we get 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑛 𝑟+1 + 𝑚 𝑟+2 if 𝑟 = 3 then 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑛 4 + 𝑚 5 = 5𝑛+4𝑚 20 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛 4 ∗ 𝑚 5 = 𝑚𝑛 20 . (A) Cubic graph of 6 vertices (B) Line graph of 3-prism graph Figure 2.6.1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 681 https://internationalpubls.com iii) and iv) For upper fractional domination number we used result 2.1.2 Let 𝐺 is 𝑟-regular graph of n vertices and m edges then Г𝑓 (𝐺) ≤ 𝑛/𝑟. So The generalized result for connected cubic graphs with degree three at every vertex of order 𝑛 = {4,6,8, … } and size 𝑚 for 𝑟 = 3, we get Г𝑓 (𝐺) ≤ 𝑛 3 and 𝐿(𝐺) is (𝑟 + 1) regular graph so Г𝑓 (𝐿(𝐺)) ≤ 𝑚 𝑟+1 we get Г𝑓 (𝐿(𝐺)) ≤ 𝑚 4 . So sum Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑛 3 + 𝑚 4 = 4𝑛+3𝑚 12 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑛 3 ∗ 𝑚 4 = 𝑚𝑛 12 . □ 2.7 Cartesian product of graph (𝑲𝟐 × 𝑷𝒏 ) If we take Path graph 𝑃𝑛 and Complete graph 𝐾2 then Cartesian product graph by 𝐾2 × 𝑃𝑛 . For the Complete graph like 𝐾2 we have the set of vertices 𝑉(𝐾2) = {𝑘1, 𝑘2} and the set of vertices of path graph 𝑃𝑛 is {𝑝1, 𝑝2,…,𝑝𝑛} respectively. The set of verticesvfor Cartesian product of 𝐾2 × 𝑃𝑛 is 𝑉 (𝐾2 × 𝑃𝑛 ) = 𝑉 (𝐾2) × 𝑉( 𝑃𝑛 ) and 𝑉(𝐾2) × 𝑉( 𝑃𝑛 ) = {(𝑘𝑖, 𝑝𝑗) | 𝑘𝑖 ∈ 𝑉(𝐾2), 𝑝𝑗 ∈ 𝑉( 𝑃𝑛 )} and ‘𝑒’ be the edge for Cartesian product 𝐾2 × 𝑃𝑛 iff 𝑒 = < (𝑘𝑖, 𝑝𝑗) , (𝑘𝑟 , 𝑝𝑠) > where the condition holds as 𝑖) 𝑖 = 𝑟 and 𝑝𝑗 𝑝𝑠 ∈ 𝐸𝑑𝑔𝑒 𝑠𝑒𝑡 𝑜𝑓 ( 𝑃𝑛 ), 𝑖𝑖) 𝑗 = 𝑠 and 𝑝𝑗 𝑝𝑠 ∈ 𝐸𝑑𝑔𝑒 𝑠𝑒𝑡 𝑜𝑓 (𝐾2). We need the following theorems on 𝛾𝑓 (𝐾2 × 𝑃𝑛 ). Theorem 2.7.1 ([10,11]) If 𝐺 is the Cartesian product graph (𝐾2 × 𝑃𝑛 ) for 𝑛 > 1 then 𝛾𝑓 (𝐾2 × 𝑃𝑛 ) = { 𝑛 + 1 2 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 𝑛2 + 2𝑛 2(𝑛 + 1) , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 Theorem 2.7.2 If 𝐺 is the Cartesian product (𝐾2 × 𝑃𝑛 ) of order 2𝑛 for 𝑛 > 1 and 𝐿(𝐺) be its line graph of order 3𝑛 − 2 then 𝛾𝑓 (𝐿(𝐾2 × 𝑃𝑛 )) = 2𝑛 3 . Proof: In Cartesian product graph (𝐾2 × 𝑃𝑛 ) for 𝑛 > 1, we noted that 𝐾2 and 𝑃𝑛 both are one distance graph and the Cartesian product of the two unit distance graphs is also unit distance graph. Let 𝐺1 = (𝐾2 × 𝑃2) is the cartesian product of 𝐾2 𝑎𝑛𝑑 𝑃2 we can see that it is cycle graph of order 4 so its fractional domination number is 𝛾𝑓 (𝐾2 × 𝑃2 ) = (2)2+2(2) 2(2+1) = 4/3 and its line graph 𝐿(𝐺1) is isomorphic to original graph 𝐺1 = (𝐾2 × 𝑃2) of order 3(2) − 2 = 4 so 𝛾𝑓 (𝐿(𝐾2 × 𝑃2 )) = 2(2) 3 = 4/3. Let 𝐺2 = (𝐾2 × 𝑃3) be the cartesian product of 𝐾2 𝑎𝑛𝑑 𝑃3 we can see that 𝛾𝑓 (𝐾2 × 𝑃3 ) = 3+1 2 = 2. Its line graph 𝐿(𝐺2) is of order 3(3) − 2 = 7 so 𝛾𝑓 (𝐿(𝐾2 × 𝑃3 )) = 2(3) 3 = 2 by assigning each vertex with weight (1/3) except central vertex. For central (hub) vertex we can assign weight zero so that adjacency relation will satisfied. Similarly if 𝐺3 = 𝐾2 × 𝑃4 be the Cartesian product of 𝐾2 𝑎𝑛𝑑 𝑃4 is graph of 8 vertices with {𝑣2, 𝑣3, 𝑣6, 𝑣7} having weights (1/5) and {𝑣1, 𝑣4, 𝑣5, 𝑣8} having weights (2/5) so 𝛾𝑓 (𝐾2 × 𝑃4 ) = Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 682 https://internationalpubls.com 42+2(4) 2(4+1) = 12 5 = 2.4 (See the Figure 2.7.2A). (A) Cartesian product of ( 𝐾2 × 𝑃4 ) (B) Its Line graph ( 𝐾2 × 𝑃4 ) Figure 2.7.2 Its line graph 𝐿(𝐺3) is of order 3(4) − 2 = 10 so 𝛾𝑓 (𝐿(𝐾2 × 𝑃4 )) = 2(4) 3 = 8/3 with vertices {𝑒1, 𝑒2, 𝑒3, 𝑒4, 𝑒5, 𝑒6, 𝑒7, 𝑒10, } having weights (1/3) and vertices {𝑒8, 𝑒9} having weights zero. (See the Figure 2.7.2B). These are minimal fractional domination numbers by considering the closed neighborhood of each vertex satisfying the condition where the vertex 𝑤 ∈ 𝑁[𝑉] such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1. Hence in general form by theorem 2.7.1 we get 𝛾𝑓 (𝐾2 × 𝑃𝑛 ) = 𝑛+1 2 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 and 𝛾𝑓 (𝐾2 × 𝑃𝑛 ) = 𝑛2+2𝑛 2(𝑛+1) , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛. Therefore in its line graph L(𝐾2 × 𝑃𝑛) with 3𝑛 − 2 vertices for 𝑛 > 1 we have 𝛾𝑓 (𝐿(𝐾2 × 𝑃𝑛 )) = 2𝑛 3 . □ 2.8 Cartesian product graph (𝑲𝟑 × 𝑷𝒏 ) If we take the path graph 𝑃𝑛 and Complete graph 𝐾3 then Cartesian product graph represented by 𝐾3 × 𝑃𝑛 . For the three vertex Complete graph 𝐾3, The vertex set 𝑉(𝐾3) = {𝑘1, 𝑘2, 𝑘3} and the vertex set for path graph 𝑃𝑛 is {𝑝1, 𝑝2,…𝑝𝑛} resp. The vertex set for Cartesian product of (𝐾3 × 𝑃𝑛 ) is 𝑉 (𝐾3 × 𝑃𝑛 ) = 𝑉(𝐾3) × 𝑉( 𝑃𝑛) and 𝑉(𝐾3) × 𝑉( 𝑃𝑛 ) = {(𝑘𝑖 , 𝑝𝑗) | 𝑘𝑖 ∈ 𝑉(𝐾3), 𝑝𝑗 ∈ 𝑉( 𝑃𝑛 )} and ‘𝑒’ be the edge for Cartesian product (𝐾3 × 𝑃𝑛 ) iff 𝑒 = < (𝑘𝑖 , 𝑝𝑗) , (𝑘𝑟 , 𝑝𝑠) > where the condition holds as 𝑖) 𝑖 = 𝑟 and 𝑝𝑗 𝑝𝑠 ∈ 𝐸𝑑𝑔𝑒 𝑠𝑒𝑡 𝑜𝑓 ( 𝑃𝑛 ), 𝑖𝑖) 𝑗 = 𝑠 and 𝑝𝑗 𝑝𝑠 ∈ 𝐸𝑑𝑔𝑒 𝑠𝑒𝑡 𝑜𝑓 (𝐾3). Theorem 2.8.1 Let G is the Cartesian product graph (𝐾3 × 𝑃𝑛 ) of order 3𝑛 for 𝑛 ≥ 3 and 𝐿(𝐺) be its line graph of order (6𝑛 − 3). then 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 3𝑛+2 5 and 𝛾𝑓 (𝐿(𝐾3 × 𝑃𝑛 )) = 3(2𝑛+1) 7 . Proof: For the Complete graph 𝐾3 and path graph 𝑃𝑛 , we have graph of Cartesian product denoted by (𝐾3 × 𝑃𝑛 ) of order 3𝑛 for 𝑛 ≥ 3. Let 𝐺1 = (𝐾3 × 𝑃3) be the Cartesian product of 𝐾3 𝑎𝑛𝑑 𝑃3 we can see that it is graph of order 9 so its fractional domination number is 𝛾𝑓 (𝐾3 × 𝑃3 ) = 3(3)+2 5 = 11/5 by assigning weight (1/5) to every vertex of the set {𝑣1, 𝑣2, 𝑣3, 𝑣5, 𝑣6, 𝑣7, 𝑣9,} and weight (2/5) to every vertex of the set {𝑣4, 𝑣8,}. For 𝐺 is Cartesian product graph (𝐾3 × 𝑃𝑛 ) of order 3𝑛 for 𝑛 ≥ 3 then for fractional domination numbers we found sequence of numbers as {11/5, 14/5, 17/5, 20/5, … 3𝑛+2 5 } so that in general 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 3𝑛+2 5 see the Figure 2.8.1A). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 683 https://internationalpubls.com In contradiction if we have 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 𝑘 and 𝑘 < 3𝑛+2 5 then adjacency relation is not satisfied with ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1 therefore there does not exist such 𝑘 where 𝑘 < 3𝑛+2 5 . Hence with minimal weights we have 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 3𝑛+2 5 . (A) Cartesian product of (𝑲𝟑 × 𝑷𝟑 ) Figure 2.8.1 In its line graph 𝐿(𝐺1) = 𝐿(𝐾3 × 𝑃3) of order 6(3) − 3 = 15 we get 𝛾𝑓 (𝐿(𝐾3 × 𝑃3 )) = 3[2(3)+1] 7 = 21 7 = 3. By assigning weight (1/7) to every vertex of the set {𝑒1, 𝑒3, 𝑒5, 𝑒6, 𝑒9, 𝑒10, , 𝑒12, 𝑒14, 𝑒15 } and weight (2/7) to every vertex of the set {𝑒2, 𝑒4, 𝑒7, 𝑒8 𝑒11, 𝑒13} see the Figure 2.8.1B). (B) Line graph of Cartesian product of ( 𝑲𝟑 × 𝑷𝟑 ) Figure 2.8.1 Let 𝐺2 = (𝐾3 × 𝑃4) be the Cartesian product of 𝐾3 𝑎𝑛𝑑 𝑃4 we can see that it is graph of order 12 so its fractional domination number is 𝛾𝑓 (𝐾3 × 𝑃4 ) = 3(4)+2 5 = 14/5 by assigning weight (1/5) to 10 number of vertices and weight (2/5) to other two vertices. In its line graph 𝐿(𝐺2) = 𝐿(𝐾3 × 𝑃4) of order [6(4) − 3] = 21 so 𝛾𝑓 (𝐿(𝐾3 × 𝑃4 )) = 3[2(4)+1] 7 = 27 7 by assigning weight (1/7) to 15 number of vertices and weight (2/7) to six number of vertices. These are minimal fractional domination numbers by definition satisfying the condition where the vertex 𝑤 ∈ 𝑁[𝑉] such that ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1. Hence if graph 𝐺 is the Cartesian product (𝐾3 × 𝑃𝑛 ) of order 3𝑛 for 𝑛 ≥ 3 then for the fractional Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 684 https://internationalpubls.com domination numbers we found sequence of numbers as {11/5, 14/5, 17/5, 20/5, … 3𝑛+2 5 } and 𝐿(𝐺) be its line graph of order (6𝑛 − 3) then fractional domination numbers formed by sequence {21/7, 27/7, 33/7, 39/7, … 3(2𝑛+1) 7 } therefore 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 3𝑛+2 5 and 𝛾𝑓 (𝐿(𝐾3 × 𝑃𝑛 )) = 3(2𝑛+1) 7 . In contradiction if we have 𝛾𝑓 (𝐿(𝐾3 × 𝑃𝑛 )) = 𝑘 and 𝑘 < 3(2𝑛+1) 7 then adjacency relation is not satisfied with ∑ 𝑓(𝑣) ⬚ 𝑣∈𝑁[𝑤] = 1 therefore there does not exist such 𝑘 where 𝑘 < 3(2𝑛+1) 7 . Hence with minimal weights we have 𝛾𝑓 (𝐿(𝐾3 × 𝑃𝑛 )) = 3(2𝑛+1) 7 . □ 2.9 The Cartesian product of two cycle graphs 𝑪𝒎 × 𝑪𝒏 Cycle graph 𝐶𝑚 and 𝐶𝑛 taken to form Cartesian product and is denoted by (𝐶𝑚 × 𝐶𝑛 ) whose vertex set is 𝑉 (𝐶𝑚 ) × 𝑉 (𝐶𝑛 ) where 𝑢 = (𝑢1, 𝑢2 ) and 𝑣 = (𝑣1, 𝑣2) are connected if 𝑢1 = 𝑣1 in 𝐶𝑚 and 𝑢2 is adjacent to 𝑣2 in 𝐶𝑛 or 𝑢1 is adjacent to 𝑣1 in 𝐶𝑚 and 𝑢2 = 𝑣2 in 𝐶𝑛. It is also known as torus grid graph 𝑇𝑚,𝑛 formed by Cartesian product of two cycle graphs 𝐶𝑚 and 𝐶𝑛. Theorem 2.9.1 Let cycle graphs 𝐶𝑚 and 𝐶𝑛 are two graphs with 𝑚 and 𝑛 vertices respectively and let 𝐺 = 𝐶𝑚 × 𝐶𝑛 is Cartesian product of two cycle graphs. Let 𝐿(𝐺) be its line graph then 𝑖) 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 + 2𝑚𝑛 7 𝑖𝑖) 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 ∗ 2𝑚𝑛 7 , 𝑖𝑖𝑖) Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 4 + 𝑚𝑛 3 𝑖𝑣) Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 4 ∗ 𝑚𝑛 3 . Proof i) and ii): We have taken reference to [7], in graphs of Cartesian product 𝐶𝑚 × 𝐶𝑛 ) the set of vertices of 𝐶𝑚 will be 𝑉 = {𝑣1 , 𝑣2 , 𝑣3 , … , 𝑣𝑚 } and the set of vertices of 𝐶𝑛 will be 𝑈 = {𝑢1 , 𝑢2 , 𝑢3 , … , 𝑢𝑚 } The graph 𝐺 has order 𝑚𝑛 and size 2𝑚𝑛 with each vertex has degree 4 or 4- regular graph. The line graph 𝐿(𝐺) has order 2𝑚𝑛 with each vertex has degree 6 or 6-regular graph. For fractional domination number we used the theorem 1.9 for following cases: Case 1: When 𝑚 and 𝑛 are even and distinct if 𝑚 = 4, 𝑛 = 6 then the graph 𝐺 = (𝐶𝑚 × 𝐶𝑛) has order 24 and size 48 with each vertex has degree 4. Hence we have 𝛾𝑓 (𝐶4 × 𝐶6 ) = 24/5 . The line graph 𝐿(𝐺) has 48 number of vertices with each vertex has degree 6. Therefore we get 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶4 × 𝐶6 ) = 48/7. Case 2: When 𝑚 is even and 𝑛 is odd if 𝑚 = 4, 𝑛 = 3 then the graph 𝐺 = (𝐶4 × 𝐶3 ) for example circulant graph has order 12 and size 24 with every vertex has degree 4. Hence 𝛾𝑓 (𝐶4 × 𝐶3 )= 12/5. The line graph 𝐺 has order 24 with each vertex has degree 6. Therefore we get 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶4 × 𝐶3 ) = 24/7. Case 3: When 𝑚 and 𝑛 are odd and equal if 𝑚 = 3, 𝑛 = 3 then the graph 𝐺 = 𝐶3 × 𝐶3 for example generalized quadrangle graph has order 9 and size 18 with each vertex has degree 4. Therefore we get 𝛾𝑓 (𝐶3 × 𝐶3 ) = 9/5. The line graph (𝐺) has order 18 with every vertex has degree Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 685 https://internationalpubls.com 6. Therefore 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶4 × 𝐶3 ) = 18/7. Case 4: When 𝑚 and 𝑛 are even and equal if 𝑚 = 4, 𝑛 = 4 then the graph 𝐺 = 𝐶4 × 𝐶4 for example tesseract graph has order 16 and size 32 with each vertex has degree 4. Hence we get 𝛾𝑓 (𝐶4 × 𝐶4 ) = 16 5 . The line graph (𝐺) has order 32 with every vertex has degree 6. Therefore 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶4 × 𝐶4 ) = 32/7. Case 5: When 𝑚 and 𝑛 𝑎𝑟𝑒 𝑜𝑑𝑑 𝑎𝑛𝑑 distinct if 𝑚 = 3, 𝑛 = 5 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝐺 = 𝐶3 × 𝐶5 ℎ𝑎𝑠 15 𝑛𝑢𝑚𝑏𝑒𝑟 of vertices and size 30 with every vertex has degree 4. Hence 𝛾𝑓 (𝐶3 × 𝐶5) = 15 5 = 3. The line graph (𝐺) has 30 number of vertices with each vertex has degree 6. Therefore 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶3 × 𝐶5 ) = 30/7. Hence in gene ra l i zed i f 𝐺 = 𝐶𝑚 × 𝐶𝑛 be 4-regular graph with order 𝑚𝑛 and graph 𝐿(𝐺) be 6-regular graph of order 2𝑚𝑛 then 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 + 2𝑚𝑛 7 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 ∗ 2𝑚𝑛 7 . Example 2.9.1 When 𝑚 = 3, 𝑛 = 4 the Cartesian product graph 𝐺 = 𝐶3 × 𝐶4 given in Figure 2.9.1A). So 𝛾𝑓 (𝐶3 × 𝐶4 ) = 12/5. (A) Cartesian product graph 𝐺 = 𝐶3 × 𝐶4 (A) Line graph of Cartesian product 𝐺 = 𝐶3 × 𝐶4 Figure 2.9.1 The line graph of Cartesian product 𝐺 = 𝐶3 × 𝐶4 given in Figure 2.9.1B) The line graph 𝐿(𝐺) has 24 vertices with each vertex has degree 6. Therefore 𝛾𝑓 (𝐿(𝐺)) = 𝛾𝑓 (𝐶3 × 𝐶4 ) = 24/7. iii) and iv) The theorem 2.1.2 give 𝐺 = 𝐶𝑚 × 𝐶𝑛 be 4-regular graph of order 𝑚𝑛 then its upper fractional domination number is Г𝑓 (𝐶𝑚 × 𝐶𝑛 ) ≤ 𝑚𝑛/𝑟 and 𝐿(𝐺) be 6-regular graph of order 2𝑚𝑛 then Г𝑓 (𝐿(𝐶𝑚 × 𝐶𝑛 )) ≤ 2𝑚𝑛 𝑟+2 hence Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 𝑟 + 2𝑚𝑛 𝑟+2 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 4 ∗ 𝑚𝑛 3 . □ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 686 https://internationalpubls.com 3 Applications Join and Union The first graph 𝐺 and second graph 𝐻 have two set of vertices as 𝑉1 and 𝑉2 which lack coherence or disjoint whereas the edge sets are 𝑋1 and 𝑋2 respectively. The union (𝐺 𝘜 𝐻) represented as graphs such that 𝑉 = ( 𝑉1 Ս 𝑉2) and 𝑋 = ( 𝑋1 Ս 𝑋2) If graph 𝐺 has order 𝑛1 and size 𝑚1, graph 𝐻 has order 𝑛2 and size 𝑚2 then for union 𝐺 𝘜 𝐻 we have 𝑛1+ 𝑛2 vertices and 𝑚1+ 𝑚2 edges. So It is recognized that 𝛾𝑓 (𝐺 𝘜 𝐻 ) = 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐻) and Г𝑓 (𝐺 𝘜 𝐻 ) = Г𝑓 (𝐺) + Г𝑓 (𝐻) here we have taken graph (𝐶3 𝘜 𝑃3 ) so immediately we have 𝛾𝑓 (𝐶3 𝘜 𝑃3 ) = 𝛾𝑓 (𝐶3) + 𝛾𝑓 (𝑃3) and Г𝑓 (𝐶3 𝘜 𝑃3 ) = Г𝑓 (𝐶3) + Г𝑓 (𝑃3). (see Figure 3A). The Join for graph G and H denoted by (𝐺 + 𝐻) has vertex set V (𝐺 + 𝐻) = 𝑉(𝐺) ∪ 𝑉(𝐻) and edge set 𝐸(𝐺 + 𝐻) = 𝐸(𝐺) ∪ 𝐸(𝐻) ∪ {(𝑔ℎ)|𝑔 ∈ 𝑉(𝐺), ℎ ∈ 𝑉(𝐻)}. If graph 𝐺 has order 𝑛1 and size 𝑚1, graph 𝐻 has order 𝑛2 and 𝑚2 number of edges then for join 𝐺 + 𝐻 we have 𝑛1 + 𝑛2 vertices and (𝑚1 + 𝑚2 + 𝑛1𝑛2) number of edges. Here we consider the graph (𝐶3 + 𝑃3) (see Figure 3B). (A) Graph 𝐶3 U 𝑃3 (B) Graph 𝐶3 + 𝑃3 Figure 3 We will now state several results involving fractional domination of graphs. Clearly, 𝛾𝑓 (𝐺 + 𝐻) ≤ min {𝛾𝑓 (𝐺), 𝛾𝑓 (𝐻)} and Г𝑓 (𝐺 + 𝐻) = max {Г𝑓 (𝐺), Г𝑓 (𝐻)} . We have the following theorems on 𝛾𝑓 (𝐺 + 𝐻) 𝑎𝑛𝑑 Г𝑓 (𝐺 + 𝐻). Theorem 3.1 ([4]). Given graphs 𝐺 and 𝐻, we have 𝛾𝑓 (𝐺 + 𝐻) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐻) = 1, 2 − 𝛾𝑓 (𝐺)+ 𝛾𝑓 (𝐻) − 2 𝛾𝑓 (𝐺)∗ 𝛾𝑓 (𝐻) − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 . And Г𝑓 (𝐺 + 𝐻) = 2 − Г𝑓 (𝐺)+ Г𝑓 (𝐻) − 2 Г𝑓 (𝐺)∗ Г𝑓 (𝐻) − 1 . Theorem 3.2 ([4]). For graph 𝐺 and graph 𝐻 we have using 1/∞ = 0, 𝛾𝑓 (𝐺 + 𝐻) = 2 − { 1/ 𝛾𝑓 (𝐻), 𝑖𝑓 𝛾𝑓 (𝐺) = ∞ 1/ 𝛾𝑓 (𝐺), 𝑖𝑓 𝛾𝑓 (𝐻) = ∞ Г𝑓 (𝐺)+ Г𝑓 (𝐻) − 2 Г𝑓 (𝐺)∗ Г𝑓 (𝐻) − 1 , 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 687 https://internationalpubls.com (Figure 3A) as union for 3 vertex cycle and 3 vertex path graph and (Figure 3B) as join for the same. So 𝛾𝑓 (𝐶3) = 1 whereas Г𝑓 (𝐶3) = 3/2, 𝛾𝑓 (𝑃3) = 1 and Г𝑓 (𝑃3) = 3/2, Same concept for 5 vertex cycle graph and 4 vertex path graph, as per the join and union we get 𝛾𝑓 (𝐶5) = 5/3 and Г𝑓 (𝐶5) = 5/2 whereas 𝛾𝑓 (𝑃4) = 2 and Г𝑓 (𝑃4) = 4/2. Therefore theorem on union gives 𝛾𝑓 (𝐶5 𝘜 𝑃4 ) = 5/3 + 2 = 11/3 and Г𝑓 (𝐶5 𝘜 𝑃4 ) = 5/2 + 2 = 9/2. Theorem on join gives 𝛾𝑓 (𝐶5 + 𝑃4) = 2 − (5/3)+2−2 (5/3)∗2−1 = 9/7 and Г𝑓 (𝐶5 + 𝑃4) = 2 − (5/2)+2−2 (5/2)∗2−1 = 11/8. Theorem 3.3 Let 𝐺 is cycle graph 𝐶𝑛 and 𝐿(𝐺) is line graph, we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 2𝑛 3 − 2 𝑛2 9 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 𝑛 − 2 𝑛2 4 − 1 . Proof. Theorem 2.1.1 and Theorem 3.1 gives 𝛾𝑓 (𝐶𝑛 ) = 𝑛/3 and 𝛾𝑓 (𝐿(𝐶𝑛 )) = 𝑛/3 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 2𝑛 3 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛2 9 hence 𝛾𝑓 (𝐶𝑛 + 𝐿(𝐶𝑛 )) = 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1 otherwise 𝛾𝑓 (𝐶𝑛 + 𝐿(𝐶𝑛 )) = 2 − 2𝑛 3 − 2 𝑛2 9 − 1 = 2 − 6𝑛−18 𝑛2−9 . Similarly Г𝑓 (𝐶𝑛 ) = 𝑛/2 and Г𝑓 (𝐿(𝐶𝑛 )) = 𝑛/2 so Г𝑓 (𝐶𝑛 ) + Г𝑓 (𝐿(𝐶𝑛 )) = 𝑛 and Г𝑓 (𝐶𝑛 ) ∗ Г𝑓 (𝐿(𝐶𝑛 )) = 𝑛2 4 hence Г𝑓 (𝐶𝑛 + 𝐿(𝐶𝑛 )) = 2 − 𝑛 − 2 𝑛2 4 − 1 = 2 − 4𝑛 − 8 𝑛2−4 . □ Theorem 3.4 For the graph 𝐺 as complete graph 𝐾𝑛 and 𝐿(𝐺) be its line graph, we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 1 + 𝑛(𝑛−1)/2 (2𝑛−4)+1 − 2 𝑛(𝑛−1)/2 (2𝑛−4)+1 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 1 + 𝑛(𝑛−1)/2 2𝑛−4 − 2 𝑛(𝑛−1)/2 2𝑛−4 − 1 . Proof: Theorem 2.2.1 and Theorem 3.1 gives 𝛾𝑓 (𝐾𝑛 ) = 1 and 𝛾𝑓 (𝐿(𝐾𝑛 )) = 𝑛(𝑛−1)/2 (2𝑛−4)+1 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛−1)/2 (2𝑛−4)+1 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1)/2 (2𝑛−4)+1 . Hence 𝛾𝑓 (𝐾𝑛 + 𝐿(𝐾𝑛 )) = 1, 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, otherwise Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 688 https://internationalpubls.com 𝛾𝑓 (𝐾𝑛 + 𝐿(𝐾𝑛 )) = 2 − 1 + 𝑛(𝑛−1)/2 (2𝑛−4)+1 − 2 𝑛(𝑛−1)/2 (2𝑛−4)+1 − 1 . Similarly Г𝑓 (𝐾𝑛 ) = 1 and Г𝑓 (𝐿(𝐾𝑛 )) = 𝑛(𝑛−1)/2 2𝑛−4 so Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 1 + 𝑛(𝑛−1)/2 2𝑛−4 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 𝑛(𝑛−1)/2 2𝑛−4 hence Г𝑓 (𝐾𝑛 + 𝐿(𝐾𝑛 )) = 2 − 1 + 𝑛(𝑛−1) 2 2𝑛−4 − 2 𝑛(𝑛−1) 2 2𝑛−4 − 1 . □ Theorem 3.5 Let 𝐺 is the 𝑆𝑛 as star graph, 𝐿(𝐺) be its line graph, we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 𝑛 + 𝑛−1 𝑛−2 − 2 𝑛2−𝑛 𝑛−2 − 1 . Proof: Theorem 2.3.1 and Theorem 3.1 gives 𝛾𝑓 (𝑆𝑛 ) = 1 and 𝛾𝑓 (𝐿(𝑆𝑛 )) = 1 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 2 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 1 hence 𝛾𝑓 (𝑆𝑛 + 𝐿(𝑆𝑛 )) = 1 if 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1 otherwise 𝛾𝑓 (𝑆𝑛 + 𝐿(𝑆𝑛 )) = 2. Similarly Г𝑓 (𝑆𝑛 ) = 𝑛 and Г𝑓 (𝐿(𝑆𝑛 )) = 𝑛−1 𝑛−2 so Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑛 + 𝑛−1 𝑛−2 and Г𝑓 (𝐶𝑛 ) ∗ Г𝑓 (𝐿(𝐶𝑛 )) ≤ 𝑛2−𝑛 𝑛−2 hence Г𝑓 (𝑆𝑛 + 𝐿(𝑆𝑛 )) ≤ 2 − 𝑛 + 𝑛−1 𝑛−2 − 2 𝑛2−𝑛 𝑛−2 − 1 . □ Theorem 3.6 Let 𝐺 is the graph (𝐾1,𝑛 ) a as bistar graph and 𝐿(𝐺) be its line graph, we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 1 if 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1 and Г𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 2𝑛 4𝑛−1 . Proof: Theorem 2.4.1 and Theorem 3.1 gives 𝛾𝑓 (𝐺) = 2 and 𝛾𝑓 (𝐿(𝐺)) = 1 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 3 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 2 hence 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 1 if 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1. Similarly Г𝑓 (𝐺) = 2𝑛 and Г𝑓 (𝐿(𝐺)) = 2 so Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) = 2𝑛 + 2and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) = 4𝑛 , Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 2𝑛 4𝑛 − 1 . □ Theorem 3.7 For the graph 𝐺 be the wheel graph on 𝑛 vertices with size 𝑚 and 𝐿(𝐺) be its line graph, where 𝛿(𝐺) is minimum degree of 𝐿(𝐺) we have 𝛾𝑓 (𝐺 + 𝐿(𝐺))= { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 1 + 𝑚 𝛿(𝐺)+1 − 2 𝑚 𝛿(𝐺)+1 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 2 + 𝑛−1 2 − 2 𝑛−2 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 689 https://internationalpubls.com Proof: Theorem 2.5.1 and Theorem 3.1 give 𝛾𝑓 (𝑊𝑛) = 1 and 𝛾𝑓 (𝐿(𝑊𝑛)) = 𝑚 𝛿(𝐺)+1 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 1 + 𝑚 𝛿(𝐺)+1 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑚 𝛿(𝐺)+1 hence 𝛾𝑓 (𝑊𝑛 + 𝐿(𝑊𝑛)) = 1 if 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, otherwise 𝛾𝑓 (𝑊𝑛 + 𝐿(𝑊𝑛)) = 2 − 1 + 𝑚 𝛿(𝐺)+1 − 2 𝑚 𝛿(𝐺)+1 − 1 Similarly Г𝑓 (𝑊𝑛) = 2 and Г𝑓 (𝐿(𝑊𝑛)) = 𝑛 − 1/2 so Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 2 + 𝑛−1 2 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑛 − 1 hence Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 2 + 𝑛−1 2 − 2 𝑛−2 . □ Theorem 3.8 If graph 𝐺 be the connected cubic graph or trivalent graph with order n, size m and 𝐿(𝐺) be its line graph of order m we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)= { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 5𝑛+4𝑚 20 − 2 𝑚𝑛 20 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 4𝑛+3𝑚 12 − 2 𝑚𝑛 12 − 1 . Proof: Theorem 2.6.1 and Theorem 3.1 gives 𝛾𝑓 (𝐺) = 𝑛/4 and 𝛾𝑓 (𝐿(𝐺)) = 𝑚/5 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑛 4 + 𝑚 5 = 5𝑛+4𝑚 20 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑛 4 ∗ 𝑚 5 = 𝑚𝑛/20 hence 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 1 if 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1 otherwise 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 5𝑛+4𝑚 20 − 2 𝑚𝑛 20 − 1 . Similarly Г𝑓 (𝐺) = 𝑛/3 and Г𝑓 (𝐿(𝐺)) = 𝑚/4. So Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑛 3 + 𝑚 4 = 4𝑛+3𝑚 12 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛/12 hence Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 4𝑛+3𝑚 12 − 2 𝑚𝑛 12 − 1 . □ Theorem 3.9 Let 𝐺 is the Cartesian product graph (𝐾2 × 𝑃𝑛 ) for 𝑛 > 1 and 𝐿(𝐺) be line graph of order 3𝑛 − 2 we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 2 − 𝑛 + 1 2 + 2𝑛 3 − 2 𝑛 + 1 2 ∗ 2𝑛 3 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 2 − 𝑛2 + 2𝑛 2(𝑛 + 1) + 2𝑛 3 − 2 𝑛2 + 2𝑛 2(𝑛 + 1) ∗ 2𝑛 3 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛. Proof: Theorem 2.7.1 and Theorem 3.1 gives 𝛾𝑓 (𝐾2 × 𝑃𝑛 ) = 𝑛+1 2 , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 and 𝛾𝑓 (𝐾2 × 𝑃𝑛 ) = 𝑛2+2𝑛 2(𝑛+1) , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛. The theorem 2.7.2 gives 𝛾𝑓 (𝐿(𝐾2 × 𝑃𝑛 )) = 2𝑛 3 . so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑛+1 2 + 2𝑛 3 if n is odd and 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑛2+2𝑛 2(𝑛+1) + 2𝑛 3 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛. Hence Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 690 https://internationalpubls.com 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 𝑛+1 2 + 2𝑛 3 − 2 𝑛+1 2 2𝑛 3 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑 and 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 𝑛2+2𝑛 2(𝑛+1) + 2𝑛 3 − 2 𝑛2+2𝑛 2(𝑛+1) ∗ 2𝑛 3 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 . □ Theorem 3.10 Let G is the Cartesian product graph (𝐾3 × 𝑃𝑛 ) of order 3𝑛 for 𝑛 ≥ 3 and 𝐿(𝐺) is its line graph of order (6𝑛 − 3) then 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 3𝑛 + 2 5 + 3(2𝑛 + 1) 7 − 2 3𝑛 + 2 5 ∗ 3(2𝑛 + 1) 7 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Proof: Theorem 2.8.1 and Theorem 3.1 gives 𝛾𝑓 (𝐾3 × 𝑃𝑛 ) = 3𝑛+2 5 and 𝛾𝑓 (𝐿(𝐾2 × 𝑃𝑛 )) = 3(2𝑛+1) 7 so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 3𝑛+2 5 + 3(2𝑛+1) 7 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 3𝑛+2 5 ∗ 3(2𝑛+1) 7 hence 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1 and 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 3𝑛+2 5 + 3(2𝑛+1) 7 − 2 3𝑛+2 5 ∗ 3(2𝑛+1) 7 − 1 otherwise. □ Theorem 3.11 Let 𝐶𝑚 and 𝐶𝑛 are the two cycle graphs with 𝑚 and 𝑛 vertices respectively and let 𝐺 = 𝐶𝑚 × 𝐶𝑛 be the Cartesian product. Let 𝐿(𝐺) be its line graph we have 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = { 1 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, 2 − 𝑚𝑛 5 + 2𝑚𝑛 7 − 2 𝑚𝑛 5 ∗ 2𝑚𝑛 7 − 1 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 and Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 𝑚𝑛 4 + 𝑚𝑛 3 − 2 𝑚𝑛 4 ∗ 𝑚𝑛 3 − 1 Proof: Theorem 2.9.1 and Theorem 3.1 gives 𝛾𝑓 (𝐺) = 𝑚𝑛 5 and 𝛾𝑓 (𝐿(𝐺)) = 2𝑚𝑛 7 where 𝐺 = 𝐶𝑚 × 𝐶𝑛 observed as Cartesian product cycle graphs so 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 + 2𝑚𝑛 7 and 𝛾𝑓 (𝐺) ∗ 𝛾𝑓 (𝐿(𝐺)) = 𝑚𝑛 5 + 2𝑚𝑛 7 hence 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 1, 𝑖𝑓 𝛾𝑓 (𝐺) = 1 𝑜𝑟 𝛾𝑓 (𝐿(𝐺)) = 1, otherwise 𝛾𝑓 (𝐺 + 𝐿(𝐺)) = 2 − 𝑚𝑛 5 + 2𝑚𝑛 7 − 2 𝑚𝑛 5 ∗ 2𝑚𝑛 7 − 1 . Similarly Г𝑓 (𝐺) ≤ 𝑚𝑛 4 and Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 3 so Г𝑓 (𝐺) + Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 4 + 𝑚𝑛 3 and Г𝑓 (𝐺) ∗ Г𝑓 (𝐿(𝐺)) ≤ 𝑚𝑛 4 ∗ 𝑚𝑛 3 hence Г𝑓 (𝐺 + 𝐿(𝐺)) ≤ 2 − 𝑚𝑛 4 + 𝑚𝑛 3 − 2 𝑚𝑛 4 ∗ 𝑚𝑛 3 − 1 . 4. Applications • Transportation Networks: Graph theory helps in optimizing transportation networks, such as road networks or flight routes. Line graphs can represent these networks where nodes represent intersections or airports, and edges represent roads or flight paths. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 691 https://internationalpubls.com • Communication networks: Graph theory is used in designing and optimizing communication networks, such as telephone networks or internet routing. Line graphs can represent these networks where nodes are switching points or routers, and edges represent communication links. • Computational Biology: [13] Here we have used these concepts in computational biological systems such as gene regulatory networks, protein-protein interaction networks and healthcare network optimization. Line graphs can represent these networks where nodes are biological entities and edges represent interactions or biochemical reactions. 5. Conclusion we have obtained bounds like lower and upper for sum of 𝛾𝑓 (𝐺) + 𝛾𝑓 (𝐿(𝐺)) and exploring the graphs for upper fractional domination number including cycle, complete, star, Bi-star, wheel, cubic graph, graph of Cartesian product like (𝐾2 × 𝑃𝑛 ), (𝐾3 × 𝑃𝑛 ) and (𝐶𝑚 × 𝐶𝑛). For these graph classes, the fractional domination number of 𝐿(𝐺) can be related to the fractional domination number of 𝐺. Our results show that there is a correlation between the sum of fractional domination number of a graph and its line graph. Parameters related to fractional domination in line graphs towards generalization. References [1] S. Arumugam, V. Mathew, and K. Karuppasamy, Fractional distance domination in graphs, Discussiones Mathematicae Graph Theory, 32 (2012), 449–459. DOI: /10.7151/dmgt.1609. [2] G. A. Cheston, G. Fricke, S.T. Hedetniemi, and D. P. Jacobs, On the computational complexity of upper fractional domination, Discrete Applied Mathematics, 27 (1990), 195-20. DOI: /10.1016/0166-218X(90)90065-K. [3] G. S. Domke, S.T. Hedetniemi, and R. C. Laskar, Fractional packings, coverings and irredundance in graphs, Congressus Numerantium. Published (1988), 66, 227-238. [4] D.C. Fisher, Fractional domination and fractional total domination of graph complements, Discrete Applied Mathematics, 122 (2002), 283-291, DOI: /10.1016/S0166-218X(01)00305-5. [5] T.W. Haynes S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, (1998). [6] T.W. Haynes , S.T. Hedetniemi , and P.J. Slater, Domination in graphs advanced topics, Pure and applied mathematics, CRC press Taylor & Francis group, Marcel Dekker, Inc., New York, (1998). [7] Frank Harary, Graph theory, Addison-Wesley publishing company Inc, (1969). [8] E. Murugan, and J. P. Joseph, On the domination number of graph and its line graph, International Journal of Mathematical Combinatorics, ISSN 1937-1055, 1 (2018), 170-181. [9] E. Murugan and J. P. Joseph, Further Results on Domination Number of a Graph and Its Line Graph, Global Journal of Engineering Science and Researches, ISSN 2348 – 8034, 6 (4), (2019), 222-229, DOI: /10.5281/zenodo.2649083. [10] E. R. Scheinerman, and D.H. Ullman, Fractional Graph Theory, Centre National de la Recherche Scientifique Paris, France, John Wiley & Sons, (2008). [11] M. Sarada, R. Jain, and G. Mundhe, Bounds for fractional domination number of some graphs and their dual graphs, Indian Journal of Natural Sciences, 14 (80), (2023), 61661-61670. ISSN: 0976 – 0997. [12] P. S. Gholap and V. E. Nikumbh, Topological spaces generated by graph, Jñānābha, 52(1) (2022), 1-7. [13] M. Sarada, R. Jain, and G. Mundhe, Applications of the Fractional Domination in Computational Biology using LPP Formulation, African Journal of Biological Sciences, 06 (05), (2024), ISSN: 2663-2187, 4341-4358. https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Teresa+W.+Haynes&search-alias=stripbooks https://www.amazon.in/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=Stephen+Hedetniemi&search-alias=stripbooks https://www.amazon.in/s/ref=dp_byline_sr_book_3?ie=UTF8&field-author=Peter+Slater&search-alias=stripbooks