International Journal of Analysis and Applications Volume 16, Number 6 (2018), 809-821 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-809 ALGEBRAIC STRUCTURE OF GRAPH OPERATIONS IN TERMS OF DEGREE SEQUENCES VISHNU NARAYAN MISHRA1,2∗, SADIK DELEN3 AND ISMAIL NACI CANGUL3 1Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur 484 887, Madhya Pradesh, India 2L. 1627 Awadh Puri Colony Beniganj, (Phase-III), Opposite - I.T.I., Ayodhya Main Road, Faizabad 224 001, Uttar Pradesh, India 3Dept. of Mathematics, Uludag University, Gorukle 16059 Bursa, Turkey ∗Corresponding author vishnunarayanmishra@gmail.com Abstract. In this paper, by means of the degree sequences (DS) of graphs and some graph theoretical and combinatorial methods, we determine the algebraic structure of the set of simple connected graphs according to two graph operations, namely join and Corona product. We shall conclude that in the case of join product, the set of graphs forms an abelian monoid whereas in the case of Corona product, this set is not even associative, it only satisfies two conditions, closeness and identity element. We also give a result on distributive law related to these two operations. 1. Introduction 1,2 Let G = (V (G),E(G)) be a simple and connected graph with | V (G) |= n vertices and | E(G) |= m edges. Usually we use the notations V and E instead of V (G) and E(G), respectively. Here, by the word ”simple”, we mean that the graphs we consider do not have loops or multiple edges. Similar studies can be Received 2017-11-27; accepted 2018-02-01; published 2018-11-02. 2010 Mathematics Subject Classification. 05C07, 05C10, 05C30, 05C76. Key words and phrases. graph; degree sequence; join; Corona product; graph operation. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 809 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-809 Int. J. Anal. Appl. 16 (6) (2018) 810 done for non-simple graphs as well. For a vertex v ∈ V , we denote the degree of v by dG(v), which is defined as the number of edges of G meeting at v. A vertex with degree one is called a pendant vertex. The notion of degree of a graph provides us an area to study various structural properties of graphs and hence attracts the attention of many graph theorists and also other scientists including chemists. If di, 1 ≤ i ≤ n, are the degrees of the vertices vi of a graph G in any order, then the degree sequence (DS) of G is the sequence {d1, d2, · · · , dn}. Also, in many papers, the DS is taken to be a non-decreasing sequence, whenever possible. Conversely, a non-negative sequence {d1, d2, · · · , dn} will be called realizable if it is the DS of any graph. It is clear from the definition that for a realizable DS, there is at least one graph having this DS. For example, the completely different two graphs in Figure 1 have the same DS. Fig. 1 Graphs with the same DS For convenience and brevity, we shall denote the DS having repeated degrees with a shorter DS. For exam- ple, if the degree di of the vertex vi appears zi times in the DS of a graph G, then we use { d (z1) 1 , d (z2) 2 , · · · , d (z`) l } instead of {d1, d2, · · · , dn} where ` ≤ n. Here the members zi are called the frequencies of the degrees. When ` = n, that is, when all degrees are different, the DS is called perfect. It is an open problem to determine that which DSs are realizable and there are several algorithms to determine that. As usual, we denote by Pn, Cn, Sn, Kn, Tr,s and Kr,s the path, cycle, star, complete, complete bipartite and tadpole graphs, respectively, which are the most used graph examples in literature, see Figure 2. Int. J. Anal. Appl. 16 (6) (2018) 811 Fig. 2 P5, C6, S7, K6, T3,2, K2,5 The number of vertices and edges of these well-known graph classes are given in Table 1. Table 1. The number of vertices and edges of some graphs G ] vertices ] edges Pn n n-1 Cn n n Sn n n-1 Kn n ( n 2 ) Kr,s r+s rs Tr,s r+s r+s Another important reason to study the DSs of graphs is topological indices. A topological index (or a graph invariant) is a fixed invariant number for two isomorphic graphs and gives some information about the graph under consideration. These indices are especially useful in the study of molecular graphs. Some of the topological indices are defined by means of the vertex degrees: first and second Zagreb indices, first and second multiplicative Zagreb indices, atom-bond connectivity index, Narumi-Katayama index, geometric- arithmetic index, harmonic index and sum-connectivity index etc. Therefore to know about the DS of the graph will help to obtain information about, e.g., the chemical properties of the graph. There are many papers on degree based topological indices, see e.g. [2]- [3]. Int. J. Anal. Appl. 16 (6) (2018) 812 The modern study of DSs started in 1981 by Bollobas, [1]. The same year, Tyshkevich et.al. established a correspondence between DS of a graph and some structural properties of this graph, [8]. In 1987, Tychke- vich et.al. written a survey on the same correspondence, [9]. In [10], the authors gave a new version of the Erdös-Gallai theorem on the realizability of a given DS. In 2008, a new criterion on the same problem is given by Triphati and Tyagi, [7]. The same year, Kim et.al. gave a necessary and sufficient condition for the same problem, [5]. Ivanyi et.al, [4], gave an enumeration of DSs of simple graphs. Miller, [6] also gave a criteria for the realizability of given DSs. There are several graph operations used in calculating some chemical invariants of graphs. Amongst these the join, cartesian, Corona product, union, disjunction, and symmetric difference are well-known. In this paper, after recalling two of these operations, join and Corona product, we shall determine the DS of these new product graphs and by means of these calculations, we shall study the algebraic properties of the join and Corona product of two graphs. Let G1 and G2 be two graphs with n1 and n2 vertices and m1 and m2 edges, respectively. The join G1∨G2 of graphs G1 and G2 with disjoint vertex sets V (G1) and V (G2) and edge sets E(G1) and E(G2) is the graph union G1 ∪G2 together with all the edges joining V (G1) and V (G2). Thus, for example, Kp ∨Kq = Kp,q, the complete bipartite graph. We have |V (G1 ∨G2)| = n1 + n2 and |E(G1 ∨G2)| = m1 + m2 + n1n2. The Corona product G1 ◦G2 of two graphs G1 and G2 is defined to be the graph Γ obtained by taking one copy of G1 (which has n1 vertices) and n1 copies of G2, and then joining the i−th vertex of G1 to every vertex in the i−th copy of G2, for i = 1, 2, · · · , n1. Let G1 = (V1,E1) and G2 = (V2,E2) be two graphs such that V (G1) = {u1, u2, · · · , un1}, |E(G1)| = m1 and V (G2) = {v1, v2, · · ·vn2}, |E(G2)| = m2. Then it follows from the definition of the Corona product that G1 ◦G2 has n1(1 + n2) vertices and m1 + n1m2 + n1n2 edges, where V (G1 ◦G2) = {(ui,vj), i = 1, 2, ...,n1; j = 0, 1, 2, ...,n2} and E(G1 ◦G2) = {((ui,v0), (uk,v0)), (ui,uk) ∈ E(G1)} ∪{((ui,vj), (ui,v`)), (vj,v`) ∈ E(G2), i = 1, 2, ...,n1} ∪{((ui,v0), (ui,v`)),` = 1, 2, ...,n2, i = 1, 2, ...,n1} . Int. J. Anal. Appl. 16 (6) (2018) 813 It is clear that if G1 is connected, then G1 ◦ G2 is connected, and in general G1 ◦ G2 is not isomorphic to G2 ◦G1. 2. Algebraic Properties of Join In this section, we deal with some algebraic properties of the join of two graphs. We shall try to determine the abstract algebraic structure of this new graph and also give the DS of the join graph G1 ∨ G2 of two graphs G1 and G2 where G1 and G2 are choosen from Pn, Cn, Sn, Kn, Tr,s and Kr,s. In particular, in the case of join operation, the set of graphs forms an abelian monoid whereas in the case of Corona product, the set of graphs is not even associative, it only satisfies two conditions, closedness and identity element. It is clear to see that there are no zero divisors for both products. Theorem 2.1. The DSs of all possible joins of the path, cycle, star, complete, tadpole and complete bipartite graphs are given in Table 2. Proof. We make the proof only for Pr ∨Ps and Sr ∨Cs. Let Pr = { 1(2), 2(r−2) } and Ps = { 1(2), 2(s−2) } . To visualize the situation, see Figure 3. There are two types of vertices in each of Pr and Ps. Therefore there are 2 + 2 = 4 types of vertices in Pr ∨ Ps. The first type is the two end vertices of Pr (red ones) which are connected with the next green vertex in Pr and s vertices in Ps. Each of these two vertices add s + 1 to the DS of Pr ∨Ps. Therefore they add (s + 1)(2). The second type of vertices are the mid ones in Pr (green ones) each of which is connected to two neigh- boring vertices in Pr and s vertices in Ps. Each of these r − 2 vertices adds s + 2 to the DS of Pr ∨ Ps. Therefore (s + 2)(r−2) is added. The third type of vertices are the two end vertices of Ps (black ones) each of which is connected to one vertex in Ps and r vertices in Pr. They add (r + 1) 2 to the DS of Pr ∨ Ps. The fourth and last type of vertices are the mid-vertices (blue ones) in Ps. Their number is s−2 and each of which similarly adds r + 2 to the DS of Pr ∨Ps. So their contribution is (r + 2)(s−2). Therefore the required DS is Pr ∨Ps = {(s + 1)(2), (s + 2)(r−2), (r + 1)(2), (r + 2)(s−2)}. Now we recall that Sr = {1(r−1),r − 1} and Cs = { 2(s) } . In Figure 4, the join of these two graphs is drawn for r = s = 5. Int. J. Anal. Appl. 16 (6) (2018) 814 Table 2. The DSs of the join of some well-known graph types G1 G2 G1 ∨ G2 Pr Ps { (s + 1)(2), (s + 2)(r−2), (r + 1)(2), (r + 2)(s−2) } Pr Cs { (s + 1)(2), (s + 2)(r−2), (r + 2)(s) } Pr Ss { (s + 1)(2), (s + 2)(r−2), (r + 1)(s−1), r + s − 1 } Pr Ks { (s + 1)(2), (s + 2)(r−2), (r + s − 1)(s) } Pr Ts,t { (s + t + 1)(2), (s + t + 2)(r−2), r + 1, (r + 2)(s+t−2), r + 3 } Pr Ks,t { (s + t + 1)(2), (s + t + 2)(r−2), (r + s)(t), (r + t)(s) } Cr Ps { (s + 2)(r), (r + 1)(2), (r + 2)(s−2) } Cr Cs { (s + 2)(r), (r + 2)(s) } Cr Ss { (s + 2)(r), r + s − 1, (r + 1)(s−1) } Cr Ks { (s + 2)(r), (r + s − 1)(s) } Cr Ts,t { (s + t + 2)(r), r + 1, (r + 2)(s+t−2), r + 3 } Cr Ks,t { (s + t + 2)(r), (r + s)(t), (r + t)(s) } Sr Ps { (s + 1)(r−1), r + s − 1, (r + 1)(2), (r + 2)(s−2) } Sr Cs { (s + 1)(r−1), r + s − 1, (r + 2)(s) } Sr Ss { (s + 1)(r−1), (r + s − 1)(2), (r + 1)(s−1) } Sr Ks { (s + 1)(r−1), s + r − 1, (r + s − 1)(s) } Sr Ts,t { (s + t + 1)(r−1), r + s + t − 1, r + 1, (r + 2)(s+t−2), r + 3 } Sr Ks,t { (s + t + 1)(r−1), r + s + t − 1, (r + s)(t), (r + t)(s) } Kr Ps { (r + s − 1)(r), (r + 1)(2), (r + 2)(s−2) } Kr Cs { (r + s − 1)(r), (r + 2)(s) } Kr Ss { (r + s − 1)(r), (r + 1)(s−1), r + s − 1 } Kr Ks { (r + s − 1)(r+s) } Kr Ts,t { (r + s + t − 1)(r), r + 1, (r + 2)(s+t−2), r + 3 } Kr Ks,t { (r + s + t − 1)(r), (r + s)(t), (r + t)(s) } Tr,s Pt { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 1)(2), (r + s + 2)(t−2) } Tr,s Ct { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 2)(t) } Tr,s St { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 1)(t−1), r + s + t − 1 } Tr,s Kt { (t + 2)(r+s−2), t + 1, t + 3, (r + s + t − 1)(t) } Tr,s Tt,m   (t + m + 2)(r+s−2), t + m + 1, t + m + 3, (r + s + 2)(t+m−2), r + s + 1, r + s + 3   Tr,s Kt,m { (t + m + 2)(r+s−2), t + m + 1, t + m + 3, (r + s + t)(m), (r + s + m)(t) } Kr,s Pt { (r + t)(s), (s + t)(r), (r + s + 1)(2), (r + s + 2)(t−2) } Kr,s Ct { (r + t)(s), (s + t)(r), (r + s + 2)(t) } Kr,s St { (r + t)(s), (s + t)(r), (r + s + 1)(t−1), r + s + t − 1 } Kr,s Kt { (r + t)(s), (s + t)(r), (r + s + t − 1)(t) } Kr,s Tt,m { (r + t + m)(s), (s + t + m)(r), (r + s + 2)(t+m−2), r + s + 1, r + s + 3 } Kr,s Kt,m { (r + t + m)(s), (s + t + m)(r), (r + s + m)(t), (r + s + t)(m) } Int. J. Anal. Appl. 16 (6) (2018) 815 There are two types of vertices in star graph Sr and only one type in cycle Cs. Therefore there are 2 + 1 = 3 types of vertices in Sr∨Cs. The first type of those is the end vertices (black ones) of the star graph Sr. Each of these is connected to the central vertex (blue coloured) by an edge in Sr and to all s vertices in Cs. Therefore each of these vertices adds s + 1, and in total, (s + 1) (r−1) is added to the DS. The second type is the unique central vertex in Sr which is connected to r − 1 end vertices in Sr and also to all of s vertices in Cs. It adds a total of r + s− 1 to the DS. The third and final type of vertices is the s vertices in Cs (green ones). Each of them adds r + 2 to the DS. Therefore a total of (r + 2) (s) is added. Hence the DS of the required join graph is Sr ∨Cs = {(s + 1)(r−1), r + s− 1, (r + 2)(s)}. � Now we can study the algebraic structure of the set of graphs according to join operation: Theorem 2.2. Let G be the set of all simple connected graphs. Then G is an abelian monoid with the join operation. Proof. Let us have three graphs G1 = {α (β11) 11 , · · · ,α (β1`) 1` }, G2 = {α (β21) 21 , · · · ,α (β2m) 2m } and G3 = {α (β31) 31 , · · · ,α (β3n) 3n } with the number of vertices n1,n2,n3, respectively. We shall show that G with the join operation is closed, associative, commutative, has identity element but no inverse elements: First of all, the join of two simple connected graphs, by definition, is another simple connected graph, so G is closed. For associativeness, note that (G1 ∨G2) ∨G3 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21) (β21), · · · , (n1 + α2m)(β2m)}∨{α (β31) 31 ,α (β32) 32 ,α (β33) 33 } = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} and G1 ∨ (G2 ∨G3) = {α (β11) 11 , · · · ,α (β1`) 1` }∨{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (n2 + α31) (β31), · · · , (n2 + α3n)(β3n)} = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} Int. J. Anal. Appl. 16 (6) (2018) 816 = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} therefore G is associative. As G1 ∨G2 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21)(β21), · · · , (n1 + α2m)(β2m)} = {(n1 + α21)(β21), · · · , (n1 + α2m)(β2m), (n2 + α11)(β11), · · · , (n2 + α1`)(β1`)} = G2 ∨G1, the operation is commutative. Therefore, to find the identity element, one needs to find a graph Z with the property that G1 ∨ Z = G1. Let Z = {ab11 , · · · ,a bk k } and let the graph Z have c vertices. Then {α(β11)11 , · · · ,α (β1`) 1` }∨{a (b1) 1 , · · · ,a (bk) k } = {α (β11) 11 , · · · ,α (β1`) 1` } implies that {(c + α11)(β11), · · · , (c + α1`)(β1`), (n1 + a1) (b1) , · · · , (n1 + ak) (bk)} = {α(β11)11 , · · · ,α (β1`) 1` } and this is only possible when c = 0. For c = 0, we have {α(β11)11 , · · · ,α (β1`) 1` , (n1 + a1) (b1) , · · · , (n1 + ak) (bk)} = {α(β11)11 , · · · ,α (β1`) 1` }. To have this equality, we must have (n1 + a1) (b1) = 0, · · · , (n1 + ak) (bk) = 0. Hence we must have no terms (n1 + a1) (b1) , · · · , (n1 + ak) (bk) in the DS of the identity element Z. This implies that b1 = · · · = bk = 0. Therefore we can symbolically take Z = {1(0)} as the identity element. Finally, let the inverse element of the graph G1 = {α (β11) 11 , · · · ,α (β1`) 1` } be denoted by {c (d1) 1 , ...,c (dk) k }. Let the number of vertices of the inverse element be e. Then G1 ∨{c (d1) 1 , · · · ,c (dk) k } = Z = {1 (0)} implies that {α(β11)11 , · · · ,α (β1`) 1` }∨{c (d1) 1 , · · · ,c (dk) k } = {1 (0)} and therefore {(e + α11)(β11), · · · , (e + α1`)(β1`), (n1 + c1) (d1) , · · · , (n1 + ck) (dk)} = {1(0)}. As there is no solution to that equation, we conclude that there is no inverse element for the join operation. Therefore the result follows. � Int. J. Anal. Appl. 16 (6) (2018) 817 3. Algebraic Properties of Corona Product Theorem 3.1. The DSs of all possible Corona products of the path, cycle, star, complete, tadpole and com- plete bipartite graphs are given in Table 3. Theorem 3.2. Let G be the set of all simple connected graphs. Then G with Corona product operation is closed with identity. Proof. First, by the definition of the operation, G is closed. Secondly, for associativeness, we should note that (G1 ◦G2) ◦G3 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (1 + α21) (n1β21), · · · , (1 + α2m)(n1β2m)}◦{α (β31) 31 , · · · ,α (β3n) 3n } = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (1 + α31) (n1(n1+n2)β31), · · · , (1 + α3n)(n1(n1+n2)β3n)} and G1 ◦ (G2 ◦G3) = { α (β11) 11 , · · · ,α (β1`) 1` } ◦{(n3 + α21)(β21), · · · , (n3 + α2m)β2m, (1 + α31) (n2β31), · · · , (1 + α3n)(n2β3n)} = {(n2 + n2n3 + α11)(β11), · · · , (n2 + n2n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (2 + α31) (n1n2β31), · · · , (2 + α3n)(n1n2β3n)}. That is, G is not associative. For the identity element, we should find a graph Z such that G1 ◦ Z = G1. Let Z = {a (b1) 1 , · · · ,a (bk) k } and let the number of vertices of Z be c. We have {α(β11)11 , · · · ,α (β1`) 1` }◦{a (b1) 1 , · · · ,a (bk) k } = {α (β11) 11 , · · · ,α (β1`) 1` } and hence, we get {(c + α11)(β11), · · · , (c + α1`)(β1`), (1 + a1) (n1b1) , · · · , (1 + ak) (n1bk)} = {α(β11)11 , · · · ,α (β1`) 1` }. For this equation to have a solution, we must have c = 0 and also b1, · · · ,bk = 0. For the sake of brevity, if we take 1 instead of 1 + a1, · · · , 1 + ak, we conclude that Z = {1(0)} is the required identity element. Let Int. J. Anal. Appl. 16 (6) (2018) 818 Table 3. The DSs of the Corona product of well-known graph types G1 G2 G1 ◦ G2 G1 G2 G1 ◦ G2 Pr Ps { 2(2r), 3(r(s−2)), (s + 1)(2), (s + 2)(r−2) } Pr Cs { 3(rs), (s + 1)(2), (s + 2)(r−2) } Pr Ss { 2(r(s−1)), s(r), (s + 1)(2), (s + 2)(r−2) } Pr Ks { s(rs), (s + 1)(2), (s + 2)(r−2) } Pr Ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 1)(2), (s + t + 2)(r−2) } Pr Ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 1)(2), (s + t + 2)(r−2) } Cr Ps { 2(2r), 3(r(s−2)), (s + 2)(r) } Cr Cs { 3(rs), (s + 2)(r) } Cr Ss { 2(r(s−1)), s(r), (s + 2)(r) } Cr Ks { s(rs), (s + 2)(r) } Cr Ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 2)(r) } Cr Ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 2)(r) } Sr Ps { 2(2r), 3(r(s−2)), (s + 1)(r−1), r + s − 1 } Sr Cs { 3(rs), (s + 1)(r−1), r + s − 1 } Sr Ss { 2(r(s−1)), s(r), (s + 1)(r−1), r + s − 1 } Sr Ks { s(rs), (s + 1)(r−1), r + s − 1 } Sr Ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 1)(r−1), r + s + t − 1 } Sr Ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 1)(r−1), r + s + t − 1 } Kr Ps { 2(2r), 3(r(s−2)), (r + s − 1)(r) } Kr Cs { 3(rs), (r + s − 1)(r) } Kr Ss { 2(r(s−1)), s(r), (r + s − 1)(r) } Kr Ks { s(rs), (r + s − 1)(r) } Kr Ts,t { 2(r), 3(r(s+t−2)), 4(r), (r + s + t − 1)(r) } Kr Ks,t { (s + 1)(rt), (t + 1)(rs), (r + s + t − 1)(r) } Tr,s Pt { 2(2(r+s)), 3((r+s)(t−2)), t + 1, (t + 2)(r+s−2), t + 3 } Tr,s Ct { 3((r+s)t), t + 1, (t + 2)(r+s−2), t + 3 } Tr,s St { 2((r+s)(t−1)), t(r+s), t + 1, (t + 2)(r+s−2), t + 3 } Tr,s Kt { t((r+s)t), t + 1, (t + 2)(r+s−2), t + 3 } Tr,s Tt,m   2(r+s), 3((r+s)(t+m−2)), 4(r+s), t + m + 1, (t + m + 2)(r+s−2), t + m + 3   Tr,s Kt,m   (t + 1)((r+s)m), (m + 1)((r+s)t), t + m + 1, (t + m + 2)(r+s−2), t + m + 3   Kr,s Pt { 2(2(r+s)), 3((r+s)(t−2)), (r + t)(s), (s + t)(r) } Kr,s Ct { 3((r+s)t), (r + t)(s), (s + t)(r) } Kr,s St { 2((r+s)(t−1)), t(r+s), (r + t)(s), (s + t)(r) } Kr,s Kt { t((r+s)t), (r + t)(s), (s + t)(r) } Kr,s Tt,m { 2(r+s), 3((r+s)(t+m−2)), 4(r+s), (r + t + m)(s), (s + t + m)(r) } Kr,s Kt,m { (t + 1)((r+s)m), (m + 1)((r+s)t), (r + t + m)(s), (s + t + m)(r) } Int. J. Anal. Appl. 16 (6) (2018) 819 T = {c(d1)1 , · · · , c (dk) k } be the inverse element of a graph G1 = {α (β11) 11 , · · · , α (β1`) 1` }. Then they must satisfy the equation G1 ◦{c (d1) 1 , · · · ,c (dk) k } = Z = {1 (0)}. If the number of vertices of T is e, then we have {α(β11)11 , · · · ,α (β1`) 1` }◦{c (d1) 1 , · · · ,c (dk) k } = {1 (0)}. In this case, this equation cannot be hold, implying that there is no inverse element in G. Finally, as {α(β11)11 , · · · ,α (β1`) 1` }◦{α (β21) 21 , · · · ,α β2m 2m } = {(n2 + α11) (β11), · · · , (n2 + α1`)(β1`), (1 + α21) (n1β21), · · · , (1 + α2m)(n1β2m)} and {α(β21)21 , · · · ,α (β2m) 2m }◦{α (β11) 11 , · · · ,α (β1`) 1` } = {(n1 + α21) (β21), · · · , (n1 + α2m)(β2m), (1 + α11) (n2β11), · · · , (1 + α1`)(n2β1`)}, G can only be commutative when G1 = G2. In general, G is not commutative. � Finally, we check whether the distributive law holds when we have join and Corona in place of · and +, or vice versa: Theorem 3.3. Neither join nor Corona operation is not distributive on each other. That is (i) G1 ∨ (G2 ◦G3) 6= (G1 ∨G2) ◦ (G1 ∨G3) , (ii) G1 ◦ (G2 ∨G3) 6= (G1 ◦G2) ∨ (G1 ◦G3). Proof. Both claims follow after the following calculations: (i) G1 ∨ (G2 ◦G3) = {α (β11) 11 , · · · ,α (β1`) 1` }∨{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (1 + α31) (n2β31), · · · , (1 + α3n)(n2β3n)} = {(n2 + n2n3 + α11)(β11), · · · , (n2 + n2n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (1 + n1 + α31) (n2β31), · · · , (1 + n1 + α3n)(n2β3n)} Int. J. Anal. Appl. 16 (6) (2018) 820 and (G1 ∨G2) ◦ (G1 ∨G3) = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21)(β21), · · · , (n1 + α2m) (β2m)}◦{(n3 + α11)(β11), · · · , (n3 + α1`)(β1`), (n1 + α31) (β31), · · · , (n1 + α3n)(β3n)} = {(n1 + n2 + n3 + α11)(β11), · · · , (n1 + n2 + n3 + α1`)(β1`), (2n1 + n3 + α21) (β21), · · · , (2n1 + n3 + α2m)(β2m), (1 + n3 + α11) (n1+n2)β11, · · · , (1 + n3 + α1`)((n1+n2)β1`), (1 + n1 + α31) ((n1+n2)β31), · · · , (1 + n1 + α3n)((n1+n2)β3n)}. (ii)G1 ◦ (G2 ∨G3) = {α (β11) 11 , · · · , α (β1`) 1` }◦{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (n2 + α31) (β31), · · · , (n2 + α3n)(β3n)} = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (1 + n2 + α31) (n1β31), · · · , (1 + n2 + α3n)(n1β3n)} and also (G1 ◦G2) ∨ (G1 ◦G3) = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (1 + α21)(n1β21), · · · , (1 + α2m) (n1β2m)}∨{(n3 + α11)(β11), · · · , (n3 + α1`)(β1`), (1 + α31) (n1β31), · · · , (1 + α3n)(n1β3n)} = {(n1 + n1n3 + n2 + α11)β11, · · · , (n1 + n1n3 + n2 + α1`)(β1`), (1 + n1 + n1n3 + α21) (n1β21), · · · , (1 + n1 + n1n3 + α2m)(n1β2m), (n1 + n1n2 + n3 + α11) β11, · · · , (n1 + n1n2 + n3 + α1`)(β1`), (1 + n1 + n1n2 + α31) (n1β31), · · · , (1 + n1 + n1n2 + α3n)(n1β3n)}. � Int. J. Anal. Appl. 16 (6) (2018) 821 References [1] B. Bollobas, Degree Sequences of Random Graphs, Discrete Math. 33 (1981), 1-19. [2] K. C. Das, N. Akgunes, M. Togan, A. Yurttas, I. N. Cangul, A. S. 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