Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 444 https://internationalpubls.com Laplacian Minimum Dominating Quotient Energy of a Graph Ramesha M S1, Purushothama S2 and Puttaswamy3 1Department of Mathematics, Government First Grade College, Kuvempunagara Mysore, Karanataka, India E-mail: profmsr1978@gmail.com 2Department of Mathematics, Maharaja Institute of Technology Mysore, Mandya, Karanataka, India Corresponding author: E-mail: psmandya@gmail.com 3Department of Mathematics, P.E.S. College of Engineering, Mandya, Karanataka, India. E-mail: prof.puttaswamy@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this paper, we present the idea of Laplacian minimum dominating quotient energy of graph, 𝐿𝑄𝐷𝐸(𝐺) and compute the Laplacian minimum dominating quotient energy of 𝐿𝑄𝐷𝐸(𝐺) of few families of graphs. Additionally, we derive bounds for the Laplacian minimum dominating quotient energy, providing a comprehensive understanding of its behavior and properties in different graph structures. Objectives: Finding the Laplacian minimum dominating quotient energy of different graph Methods: To establish the upper and lower bounds for the energy of graphs we employ the Standard methods of proofs namely direct methods and using Matlab to compute the minimum pendant dominating partition eigen values of a graph 𝐺. Results: We obtain the Laplacian minimum dominating quotient energy of 𝐿𝑄𝐷𝐸(𝐺) of well-known families of graphs. Additionally we obtain upper and lower bounds Conclusions: Nowadays, the study of theory of domination and energy of graph is an important area in Graph theory and also remarkable research is going on in this area. In recent years many scholars are working in this area and also they are introducing new domination parameters. In this paper we have initiated the study of Laplacian minimum dominating quotient energy of graph. We have calculated the energies for some standard family graphs and we have established some bounds for this parameter. Further, we have studied some important properties of Laplacian minimum quotient dominating eigenvalues Keywords: Laplacian Minimum Dominating set, Minimum Dominating Set, Quotient Energy, Laplacian Dominating Quotient Matrix. 1. Introduction Let 𝐺 = (𝑉, 𝐸) be a graph with 𝑛 nodes and π‘š edges. The degree of 𝑣𝑖 written by 𝑑(𝑣𝑖) is the number of edges incident with 𝑣𝑖. The maximum node of degree is denoted by Ξ”(𝐺) and minimum node of degree is denoted by 𝛿(𝐺). The adjacency matrix 𝐴𝐷(𝐺) of 𝐺 is defined by its entries as π‘Žπ‘–π‘— = 1 if 𝑣𝑖𝑣𝑗 ∈ 𝐸(𝐺) π‘œπ‘Ÿ 𝑣𝑖 ∈ 𝐷 𝑖𝑓 (𝑖 = 𝑗) where 𝐷 is a dominating set of 𝐺 and 0 otherwise. The eigen values of graph 𝐺 are the eigenvalues of its adjacency matrix𝐴𝐷(𝐺), denoted byπœ†1 β‰₯ πœ†2 β‰₯ β‹― β‰₯ πœ†π‘›. A graph 𝐺 is considered singular if it has at least one eigenvalue equal to zero. In the case of singular graphs, it is clear that 𝑑𝑒𝑑(𝐴) = 0. A graph is said to be nonsingular if all of its eigenvalues are nonzero. A graph 𝐺 is referred to be k-regular if every node in 𝐺 has degree π‘˜. mailto:psmandya@gmail.com mailto:prof.puttaswamy@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 445 https://internationalpubls.com The energy of a graph 𝐺 is defined as E(G) = βˆ‘ |Ξ»i|. n i=1 This concept was introduced by I. Gutman in 1978 [3]. I.Gutman and B.Zhou [4] defined the Laplacian energy of a graph 𝐺 in the year 2006. Let 𝐺 be a graph with 𝑛 nodes and π‘š edges. The Laplacian matrix of the graph 𝐺, denoted by 𝐿 = 𝐿𝑖,𝑗, is a square matrix of order 𝑛. The elements of the Laplacian matrix are defined as 𝐿𝑖𝑗 = { βˆ’1, if 𝑣𝑖 and 𝑣𝑗 are adjacent, 0, if 𝑣𝑖 and 𝑣𝑗 are non adjacent, 𝑑𝑖 𝑖𝑓 𝑖 = 𝑗. Where 𝑑𝑖 is the vertex's 𝑣𝑖 degree Let πœ†1, πœ†2, … , πœ†π‘› be the eigen values of Laplacian matrix 𝐺. Laplacian energy of 𝐺 is defined as 𝐿𝐸(𝐺) =βˆ‘| πœ†π‘– βˆ’ 2π‘š 𝑛 | 𝑛 𝑖=1 The key characteristics of Laplacian energy, including various upper and lower bounds, have been established in [4, 5] It has been found that Laplacian graph energy has notable applications in areas such as chemical analysis, high-resolution satellite image classification and segmentation, as well as identifying semantic structures in image hierarchies. In this article, we are defining a matrix, called the Laplacian minimum dominating quotient matrix denoted by 𝐿𝑄𝐷(𝐺) and we study its eigenvalues and the energy. Further, we study the mathematical aspects of the Laplacian minimum dominating quotient energy of a graph. It is possible that the Laplacian minimum dominating quotient energy discussed in this article could have uses in other fields of science, such as chemistry, and beyond. The graphs under consideration are assumed to be finite, simple, undirected, with no isolated nodes, and of order at least two. 2. Quotient Energy of Graphs For a graph 𝐺, the quotient matrix 𝑄 = 𝑄(𝐺) = π‘žπ‘–π‘— is a 𝑝 Γ— 𝑝 matrix defined as π‘žπ‘–π‘— = { 𝑑(𝑣𝑖) 𝑑(𝑣𝑗) , π‘“π‘œπ‘Ÿ 𝑣𝑖𝑣𝑗 ∈ 𝐸, 0, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. The characteristic polynomial of 𝑄(𝐺) is 𝑓(𝐺, πœ†) = 𝑑𝑒𝑑 (𝑄 βˆ’ πœ† 𝐼). The quotient spectrum of the graph 𝐺 is the eigenvalues of the matrix 𝑄 and is denoted as 𝑄 βˆ’ 𝑆𝑝𝑒𝑐(𝐺). Let πœ†1 β‰₯ πœ†2 β‰₯ . . . β‰₯ πœ†π‘› be the spectrum of 𝑄(𝐺). Then the quotient energy is defined as QE(G) = βˆ‘ |Ξ»i| n i=1 . For additional details about quotient energy of a graph refer [6] 3. Minimum Dominating Quotient Energy of Graph Let 𝐺 be simple graph of order 𝑛 with node set 𝑉 = {𝑣1, 𝑣2, . . . , 𝑣𝑛} edge set 𝐸. Let 𝐷 be the minimum dominating set of a graph 𝐺. The minimum dominating quotient matrix of 𝐺 is the 𝑛 Γ— 𝑛 matrix defined by 𝐴𝑄(𝐺) = π‘Žπ‘–π‘— where Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 446 https://internationalpubls.com π‘Žπ‘–π‘— = { 𝑑(𝑣𝑖) 𝑑(𝑣𝑗) , 𝑖𝑓 𝑣𝑖𝑣𝑗 ∈ 𝐸, 1, 𝑖𝑓 𝑣𝑖 = 𝑣𝑗 π‘Žπ‘›π‘‘ 𝑣𝑖 ∈ 𝐷, 0, 𝑖𝑓 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ . The characteristic polynomial of 𝐴𝑄(𝐺) is indicated by𝑓(𝐺, πœ†) = 𝑑𝑒𝑑 (πœ† 𝐼 βˆ’ 𝐴𝑄(𝐺)). The minimum dominating quotient eigenvalues of the graph 𝐺 are the eigenvalues of 𝐴𝑄(𝐺). Since 𝐴𝑄(𝐺) is real and symmetric, its eigenvalues are real numbers and are labelled in non-increasing order πœ†1 β‰₯ πœ†2 β‰₯ . . . β‰₯ πœ†π‘›. The minimum dominating quotient energy of 𝐺 is defined as QDE(G) =βˆ‘|Ξ»i| n i=1 4. The Laplacian Minimum Dominating Quotient Energy of a Graph Let 𝐷(𝐺) represent the diagonal matrix of the node degrees of the graph 𝐺. Then the Laplacian minimum dominating quotient matrix of 𝐺 is denoted by 𝐿𝑄𝐷𝐸(𝐺) and is defined as follows 𝐿𝑄𝐷𝐸(𝐺) = 𝐷(𝐺) βˆ’ 𝐴𝐷(𝐺). Let πœ†1 β‰₯ πœ†2 β‰₯ . . . β‰₯ πœ†π‘› be the eigen values 𝐿𝑄𝐷𝐸(𝐺) organized in non-increasing order. These eigen values are called Laplacian minimum dominating quotient eigen values of 𝐺. The Laplacian minimum dominating quotient energy of a graph 𝐺 is defined as LQDE(G) =βˆ‘|Ξ»i βˆ’ 2m n | n i=1 Where π‘š is the number of edges of 𝐺 and 2π‘š 𝑛 is the average degree of 𝐺. Example.4.1 Let 𝐺 be a graph with 6 nodes, as illustrated in Figure 4.1. The possible 𝛾 βˆ’sets are (𝑖)𝐷1 = {𝑣2, 𝑣4} (𝑖𝑖)𝐷2 = {𝑣1𝑣4} Figure. 4.1 (i) If the 𝛾 βˆ’ set is 𝐷1 = {𝑣2, 𝑣4} then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 447 https://internationalpubls.com The characteristic polynomial is expressed as 𝑓{𝑛}(𝐺, πœ†) = πœ† 7 βˆ’ 14πœ†6 + 72πœ†5 βˆ’ 166πœ†4 + 162πœ†3 βˆ’ 30πœ†2 βˆ’ 30πœ† + 4 = 0. The Laplacian minimum dominating quotient eigen values are πœ†1 = βˆ’0.5597, πœ†2 = βˆ’0.1842, πœ†3 = 0.9583, πœ†4 = 2, πœ†5 = 2.1960, πœ†6 = 4.1923, πœ†_7 = 4.3973. The mean degree of the graph 2π‘š 𝑛 = 2Γ—8 7 = 16 7 Hence, Laplacian minimum dominating quotient energy, 𝐿𝑄𝐷1𝐸(𝐺) β‰ˆ 12.43638 (ii) If the 𝛾 βˆ’ set is 𝐷2 = {𝑣1, 𝑣4} then The characteristic polynomial is expressed as 𝑓{𝑛}(𝐺, πœ†) = πœ† 7 βˆ’ 14πœ†6 + 71πœ†5 βˆ’ 155πœ†4 + 121πœ†3 + 27πœ†2 βˆ’ 48πœ† βˆ’ 4 = 0. The Laplacian minimum dominating quotient eigen values areπœ†1 = βˆ’0.4956, πœ†2 = βˆ’0.0811, πœ†3 = 1.0381, πœ†4 = 2.2342, πœ†5 = 5.09551, πœ†6 = 4.2094, πœ†7 = 2. Average degree of the graph 2π‘š 𝑛 = 2Γ—8 7 = 16 7 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 448 https://internationalpubls.com Thus, the Laplacian minimum dominating quotient energy, 𝐿𝑄𝐷2𝐸(𝐺) β‰ˆ 12.8677 Therefore, based on the above example, it is evident that the Laplacian minimum dominating quotient energy of a graph 𝐺 is influenced by the minimum dominating set of 𝐺 5. Laplacian Minimum Dominating Quotient Energy of Some Standard Graphs Theorem 5.1. If 𝐾𝑛 is the complete graph with 𝑛 nodes, then 𝐿𝑄𝐷𝐸(𝐾𝑛) = (𝑛 βˆ’ 2) + βˆšπ‘›2 βˆ’ 2𝑛 + 5. Proof: Let 𝐾𝑛 be the complete graph with node set 𝑉 = {𝑣1, 𝑣2, … , 𝑣𝑛}. The 𝛾 βˆ’set 𝐷 = {𝑣1}. And Its characteristic polynomial is [πœ† βˆ’ 𝑛](π‘›βˆ’2)[πœ†2 βˆ’ (𝑛 βˆ’ 1)πœ† βˆ’ 1] The Laplacian minimum dominating quotient eigen values are: πœ† = 𝑛[(𝑛 βˆ’ 2)π‘‘π‘–π‘šπ‘’], πœ† = (𝑛 βˆ’ 1) Β± βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 [π‘œπ‘›π‘’ π‘‘π‘–π‘šπ‘’ π‘’π‘Žπ‘β„Ž] Average degree of 𝐾𝑛 = 2π‘š 𝑛 = 2 𝑛(π‘›βˆ’1) 2 𝑛 = 𝑛 βˆ’ 1 Hence, the Laplacian minimum dominating quotient energy of 𝐾𝑛 is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 449 https://internationalpubls.com 𝐿𝑄𝐷𝐸(𝐾𝑛) = |𝑛 βˆ’ (𝑛 βˆ’ 1)| (𝑛 βˆ’ 2) + | (𝑛 βˆ’ 1) + βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 βˆ’ (𝑛 βˆ’ 1)| + | (𝑛 βˆ’ 1) βˆ’ βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 βˆ’ (𝑛 βˆ’ 1)| 𝐿𝑄𝐷𝐸(𝐾𝑛) = (𝑛 βˆ’ 2) + | βˆ’π‘› + 1 + βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 | + | βˆ’π‘› + 1 βˆ’ βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 | Therefore, 𝐿𝑄𝐷𝐸(𝐾𝑛) = (𝑛 βˆ’ 2) + βˆšπ‘›2 βˆ’ 2𝑛 + 5 Theorem. 5.2. If 𝐾1,π‘›βˆ’1 is the star graph with 𝑛 node, then 𝐿𝑄𝐷𝐸(𝐺) = (π‘›βˆ’2)2 𝑛 + βˆšπ‘›2 βˆ’ 2𝑛 + 5 Proof: Let 𝐾1,π‘›βˆ’1 be the star graph with node set 𝑉 = {𝑣1, 𝑣2… . , π‘£π‘›βˆ’1}. the minimum dominating set 𝐷 = {𝑣1} . And Its characteristic polynomial is [πœ† βˆ’ 1](π‘›βˆ’2)[πœ†2 βˆ’ (𝑛 βˆ’ 1)πœ† βˆ’ 1] The Laplacian minimum dominating quotient eigen values are: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 450 https://internationalpubls.com πœ† = 1[(𝑛 βˆ’ 2)π‘‘π‘–π‘šπ‘’], πœ† = (𝑛 βˆ’ 1) Β± βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 [π‘œπ‘›π‘’ π‘‘π‘–π‘šπ‘’ π‘’π‘Žπ‘β„Ž] Average degree of 𝐾1,π‘›βˆ’1 = 2π‘š 𝑛 = 2(π‘›βˆ’1) 𝑛 Hence, the Laplacian minimum dominating quotient energy of 𝐾1,π‘›βˆ’1 is 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1) = |𝑛 βˆ’ 2(𝑛 βˆ’ 1) 𝑛 | (𝑛 βˆ’ 2) + | (𝑛 βˆ’ 1) + βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 βˆ’ 2(𝑛 βˆ’ 1) 𝑛 | + | (𝑛 βˆ’ 1) βˆ’ βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 βˆ’ 2(𝑛 βˆ’ 1) 𝑛 | 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1) = (𝑛 βˆ’ 2)2 𝑛 + | βˆ’π‘› + 1 + βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 | + | βˆ’π‘› + 1 βˆ’ βˆšπ‘›2 βˆ’ 2𝑛 + 5 2 | Therefore, 𝐿𝑄𝐷𝐸(𝐾1,π‘›βˆ’1) = (π‘›βˆ’2)2 𝑛 + βˆšπ‘›2 βˆ’ 2𝑛 + 5 6. Bounds on Laplacian Minimum Dominating Quotient Energy of Graphs Theorem 6.1. Let 𝐷 be a minimum dominating set of a graph 𝐺 and Ξ»1, Ξ»2, . . . , Ξ»n are the eigenvalues of LQ𝐷(G) then (𝑖) βˆ‘πœ†π‘– = 2|𝐸| βˆ’ |𝐷| 𝑛 𝑖=1 (𝑖𝑖) βˆ‘πœ†π‘– 2 = 2|𝐸| +βˆ‘(𝑑𝑖 βˆ’ β„Žπ‘–) 2 𝑛 𝑖=1 𝑛 𝑖=1 π‘€β„Žπ‘’π‘Ÿπ‘’ β„Žπ‘– = { 1, 𝑖𝑓 𝑣𝑖 ∈ 𝐷 0, 𝑖𝑓 𝑣𝑖 βˆ‰ 𝐷 Proof: (i) By definition, the sum of the principal diagonal elements of LQ𝐷(G) is equal to βˆ‘di βˆ’ |D| = 2|E| βˆ’ |D| n i=1 Also, the sum of the eigenvalues of the matrix LQ𝐷(G) is equal to the traceLQ𝐷(G). (ii) The result that the sum of the squares of the eigenvalues of LQ𝐷(G) is equal to the trace of LQ𝐷(G) 2 is a direct application of a general property of matrices. Therefore, βˆ‘Ξ»i 2 = βˆ‘βˆ‘lijlji n j=1 = βˆ‘(lij) 2 n i=1 +βˆ‘(lji) 2 n j=1 n i=1 n i=1 = 2βˆ‘(lij) 2 n i