Generalization of the degree distance of the tensor product of graphs

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume , Pages 11 7 Generalization of the degree distance of the tensor product of graphs K Pattabiraman P Kandan Department of Mathematics, Faculty of Engineering and Technology Annamalai University Annamalainagar India pramank@gmailcom kandank@gmailcom Abstract In this paper, the exact formulae for the generalized degree distance, the degree distance and the reciprocal degree distance of the tensor product of a connected graph and of the complete multipartite graph with partite sets of sizes m 0, m 1,, m r 1 are obtained In addition we show that the result given by Wang and Kang [J Comb Optim, online June 014] on the degree distance of the tensor product of graphs is incorrect, and the corrected version of this result is the corollary of our main theorem 1 Introduction All the graphs considered in this paper are simple and connected For a graph G, and for vertices u, v V G, the distance between u and v in G, denoted by d G u, v, is the length of a shortest u, v-path in G We also let d G v be the degree of a vertex v V G For two simple graphs G and H, their tensor product, denoted by G H, has vertex set V G V H in which g 1, h 1 and g, h are adjacent whenever g 1 g is an edge in G and h 1 h is an edge in H Note that if G and H are connected graphs, then G H is connected only if at least one of the graphs is nonbipartite The tensor product of graphs has been extensively studied in relation to areas such as graph colorings, graph recognition, decompositions of graphs, and design theory; see [1, 3, 4, 18, 1] A topological index of a graph is a real number related to the graph; it does not depend on any labeling or pictorial representation of a graph In theoretical chemistry, molecular structure descriptors also called topological indices are used for modeling physicochemical, pharmacologic, toxicologic, biological and other properties of chemical compounds [10] There exist several types of such indices, especially those

2 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , based on vertex and edge distances One of the most intensively studied topological indices is the Wiener index Let G be a connected graph The Wiener index of G is defined as W G d G u, v, with the summation over all pairs of distinct vertices of G This 1 u, v V G definition can be further generalized in the following way: W λ G 1 u, v V G d λ Gu, v, where d λ G u, v d Gu, v λ and λ is a real number [11, 1] If λ 1, then W 1 G HG, where HG is the Harary index of G In the chemical literature both W 1 [8] as well as the general case W λ were examined [8, 13] Dobrynin and Kochetova [6] and Gutman [9] independently proposed a vertexdegree-weighted version of the Wiener index called the degree distance or the Schultz molecular topological index, which is defined for a connected graph G as DDG d G u d G vd G u, v, where d G u is the degree of the vertex u in G 1 u,v V G Note that the degree distance is a degree-weight version of the Wiener index In the literature, many results on the degree distance DDG have been put forward in past decades, and they mainly deal with extreme properties of DDG Tomescu [5] showed that the star is the unique graph with minimum degree distance within the class on n-vertex connected graphs Tomescu [6] deduced properties of graphs with minimum degree distance in the class of n-vertex connected graphs with m n 1 edges For other related results along this line, see [5, 16, 19] The additively weighted Harary indexh A or the reciprocal degree distance RDD is defined in [] as H A G RDDG 1 d G ud G v d G In [14], u,v u,v V G Hamzeh et al recently introduced the concept of the generalized degree distance of graphs Hua and Zhang [17] have obtained lower and upper bounds for the reciprocal degree distance of a graph in terms of other graph invariants, including the degree distance, Harary index, the first Zagreb index, the first Zagreb coindex, pendent vertices, independence number, chromatic number and vertex- and edgeconnectivity Pattabiraman and Vijayaragavan [, 3] have obtained the reciprocal degree distance of the join, tensor product, strong product and wreath product of two connected graphs in terms of other graph invariants The chemical applications and mathematical properties of the reciprocal degree distance are well studied in [, 0, 4] 1 The generalized degree distance, denoted by H λ G, is defined as H λ G d G u d G vd λ G u, v, where λ is a any real number If λ 1 then u,v V G H λ G DDG, and if λ 1 then H λ G RDDG The generalized degree distance of unicyclic and bicyclic graphs are studied by Hamzeh et al [14, 15] Also they have given the generalized degree distance of the Cartesian product, join, symmetric difference, composition and disjunction of two graphs It is well-known that

3 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , many graphs arise from simpler graphs via various graph operations Hence it is important to understand how certain invariants of such product graphs are related to the corresponding invariants of the original graphs In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of the tensor product G K m0, m 1,, m r 1, where K m0, m 1,, m r 1 is the complete multipartite graph with partite sets of sizes m 0, m 1,, m r 1 are obtained We show in Section that the major result proved in the paper [7] is incorrect, and the corrected version of this result is a corollary of Theorem 5 proved by us In addition, we have given a counterexample to justify our claim The first Zagreb index is defined as M 1 G d G u In fact, one can u V G rewrite the first Zagreb index as M 1 G d G ud G v Zagreb indices are uv EG found to have applications in Quantitative Structure Property Relationship QSPR and Quantitative Structure Activity Relationship QSAR studies as well; see [7] If m 0 m 1 m r 1 s in K m0, m 1,, m r 1 the complete multipartite graph with partite sets of sizes m 0, m 1,, m r 1, then we denote this graph by K rs For S V G, S denotes the subgraph of G induced by S For two subsets S, T V G, not necessarily disjoint, by the notation d G S, T we mean the sum of the distances in G from each vertex of S to every vertex of T ; that is, d G S, T d G s, t s S, t T Generalized degree distance of tensor product of graphs Let G be a connected graph with V G {v 0, v 1,, v } and let K m0, m 1,, m r 1, r 3, be the complete multiparite graph with partite sets V 0, V 1,, V r 1 with V i m i, 0 i r 1 In the graph G K m0, m 1,, m r 1, let B ij v i V j, v i V G and 0 j r 1 For our convenience, we write V G V K m0, m 1,, m r 1 i 0 i 0 { v i r 1 j 0 V j } {{v i V 0 } {v i V 1 } {v i V r 1 }} { } Bi0 B i1 B ir 1, where Bij v i V j i 0 r 1 i 0 j 0 B ij

4 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Let B {B ij } i 0,1,, We call X i r 1 B ij a layer and Y j B ij a column j 0,1,, r 1 j 0 i 0 of G K m0, m 1,, m r 1 ; see Figures 1 and Clearly, a layer respectively, column is an independent set in G K m0, m 1,, m r 1 ; in particular, B ij is an independent set Further, if v i v k EG, then the subgraph B ij B kp of G K m0, m 1,, m r 1 is isomorphic to K Vj V p or a totally disconnected graph according as or j p This is used in the proof of the next lemma v 0 vertices of K m0,m 1,,m r 1 V 0 V l V r 1 v i B il vertices of G v j v k v B kl If v i v k is on a triangle v i v j v k in G, then the distance from a vertex in B il to a vertex in B kl is and a shortest path is shown with broken edges If v i v k is an edge but not on a triangle in G, then the distance from a vertex in B il to a vertex in B kl is 3 and a shortest path is shown with solid edges Fig 1 The proof of the following lemma follows easily from the properties and structure of G K m0, m 1,, m r 1 and the paths as shown in Figures 1 and

5 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , v 0 vertices of K m0,m 1,,m r 1 V 0 V l V r 1 v i B il vertices of G v k B kl v If the distance d from v i to v k in G is even resp odd, then the distance from a vertex in B il to a vertex in B kl is d resp d and a shortest path is shown with broken resp solid edges Fig Lemma 1 Let G be a connected graph on n vertices and let B ij, B kp B of the graph G K m0, m 1,, m r 1, where r 3 i The distance between any two distinct vertices in B ij is ii The distance between any two vertices one from B ij and another from B ip,, is iii The distance between any two vertices one from B ij and another from B kj,, is or 3 according as v i v k lies on a triangle in G or v i v k EG and v i v k does not lies on a triangle in G iv If v i v k EG, then the distance between two vertices, one in B ij and the other in B kp,,, is 1

6 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , v If v i v k / EG, then the distance between two vertices, one in B ij and the other in B kp, is d G v i, v k The proof of the following lemma follows easily from Lemma 1, and hence it is left to the reader This lemma is used in the proof of the main theorem of this section Lemma Let G be a connected graph on n vertices and let B ij, B kp B of the graph G G K m0, m 1,, m r 1, where r 3 i If v i v k EG, then m j m p, if, d λ G B ij, B kp λ m j, if j p and v i v k is on a triangle of G, 3 λ m j, if j p and v i v k is not on a triangle of G { ii If v i v k / EG, then d λ G B m j m p d λ G ij, B kp v i, v k, if, m j d λ G v i, v k, if j p { iii d λ G B λ m j m j 1, if j p, ij, B ip λ m j m p, if Proof i Let v i v k EG If, then the distance between a vertex of B ij and a vertex of B kp is 1 in G and there are m j m p pairs of vertices between B ij and B kp ; hence d λ G B ij, B kp m j m p If j p and v i v k lie in a triangle respectively, not in a triangle, then the distance between a vertex of B ij and a vertex of B kj is λ respectively, 3 λ in G, and there are m j pairs of vertices between B ij and B kj ; hence d λ G B ij, B kj λ m j respectively, 3 λ m j ii If v i v k / EG, then the distance from a vertex of B ij to a vertex of B kj respectively, B kp is d λ G v i, v k in G and there are m j m p respectively, m j pair of vertices between B ij and B kp respectively, B kj ; hence d λ G B ij, B kp m j m p d λ G v i, v k respectively, d λ G B ij, B kj m jd λ G v i, v k iii The distance between a vertex of B ij and a vertex of B ip respectively, B ij is λ respectively, λ in G, and there are m j m j 1 respectively, m j m p pairs of vertices between B ij and B ip respectively, B ij ; hence d λ G B ij, B ij λ m j m j 1 respectively, d λ G B ij, B ip λ m j m p Lemma 3 Let G be a connected graph and let B ij be in G G K m0, m 1,, m r 1 Then the degree of a vertex v i, u j B ij in G is d G v i, u j d G v i n 0 m j, where n 0 r 1 m j j0

7 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Remark 4 The following sums hold: r 1 j, p 0 where n 0 r 1 m jm p n 3 0 n 0 q r 1 j0 r 1 j, p 0 j0 r 1 r 1 j, p 0 j, p 0 m j m p m j m p r 1 r 1 m p n 0 m 4 j j0 j0 and r 1 j, p 0 q, r 1 m j m 3 p, m j and q is the number of edges of K m0, m 1,, m r 1 j0 m j n 0 q, Next we determine the generalized degree distance of G K m0, m 1,, m r 1 Theorem 5 Let G be a connected graph with n vertices and m edges and let E be the set of edges of G which do not lie on any C 3 of it If n 0 and q are the numbers of vertices and edges of K m0, m 1,, m r 1, r 3, respectively, then H λ G K m0, m 1,, m r 1 n 0 q H λ G λ mqn 0 1 λ 1M 1 G 3λ λ Proof Let G G K m0, m 1,, m r 1 Clearly, H λ G 1 1 r 1 i 0 j 0 B ij, B kp B i 0 r 1 j, p 0 r 1 j 0 r 1 j, p 0 r 1 d G v i d G v k n 3 0 qn 0 d G B ij d G B kp d λ G B ij, B kp d G B ij d G B ip d λ G B ij, B ip d G B ij d G B kj d λ G B ij, B kj d G B ij d G B kp d λ G B ij, B kp d G B ij d G B ij d λ G B ij, B ij 1 {A 1 A A 3 A 4 }, 1 where A 1 to A 4 are the sums of the above terms, in order j 0

8 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , We shall calculate A 1 to A 4 of 1 separately A 1 First we compute r 1 d G B ij d G B ip i 0 j, p 0 d λ G B ij, B ip i0 i0 r 1 j, p 0 r 1 j, p 0 d G B ij d G B ip d λ G B ij, B ip n 0 m j d G v i n 0 m p d G v i λ m j m p, by Lemmas 1, and 3 r 1 λ n 0 m j m p d G v i m j m p i0 j, p 0 r 1 λ m 4n 0 q n 3 0 A Next we compute r 1 we calculate j 0 j0 d G B ij d G B kj, by Remark 4 d G B ij d G B kj d λ G B ij, B kj For this, initially d λ G B ij, B kj Let E 1 {uv EG uv is on a C 3 in G} and E EG E 1 v i v k / EG d G B ij d G B kj d λ G B ij, B kj d G B ij d G B kj d λ G B ij, B kj

9 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , v i v k E 1 v i v k / EG v i v k E 1 v i v k / EG v i v k E 1 d G B ij d G B kj d λ G B ij, B kj d G B ij d G B kj d λ G B ij, B kj n 0 m j d G v i d G v k m j d λ Gv i, v k n 0 m j d G v i d G v k λ m j n 0 m j d G v i d G v k 3 λ m j, by Lemmas and 3 n 0 m j d G v i d G v k m j d λ Gv i, v k n 0 m j d G v i d G v k λ m j m j m j n 0 m j d G v i d G v k 3 λ m j m j m j

10 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , v i v k / EG v i v k E 1 v i v k E 1 n 0 m j d G v i d G v k m j d λ Gv i, v k n 0 m j d G v i d G v k m j d λ Gv i, v k n 0 m j d G v i d G v k m j d λ Gv i, v k n 0 m j d G v i d G v k λ 1m j n 0 m j d G v i d G v k 3 λ 1m j, since d λ Gv i, v k 1 if v i v k E 1 and n 0 m j d G v i d G v k m j d λ Gv i, v k v i v k E 1 n 0 m j m j n 0 m j d G v i d G v k λ 1m j n 0 m j d G v i d G v k λ 1m j n 0 m j d G v i d G v k 3 λ λ m j H λ G M 1 G λ 1 3 λ λ d G v i d G v k, where M 1 G is the first Zagreb index of G Note that each edge v i v k of G is being counted twice in the sum, namely, v i v k and v k v i

11 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Now summing over j 0, 1,, r 1, we get r 1 j 0 r 1 d G B ij d G B kj d λ G B ij, B kj n 3 0 qn 0 H λ G λ 1M 1 G 3 λ λ by Remark 4 A 3 Next we compute r 1 j, p 0 j0 dg v i d G v k, 3 d G B ij d G B kp d λ G B ij, B kp r 1 j, p 0 r 1 j, p 0 d G B ij d G B kp d λ G B ij, B kp n 0 m j d G v i n 0 m p d G v k m j m p d λ Gv i, v k, by Lemmas and 3 n 0 m j m p m jm p d G v i d λ Gv i, v k n 0 m j m p m j m pd G v k d λ Gv i, v k r 1 j, p 0 r 1 j, p 0 n o m j m p d G v i d G v k d λ Gv i, v k m jm p d G v i d λ Gv i, v k m j m pd G v k d λ Gv i, v k, by Remark 4 r 1 qn 0 H λ G n 3 0 qn 0 H λ G r 1 4n 0 q n 3 0 j0 j0 H λ G 4

12 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , 11 7 A 4 Finally, we compute i 0 j 0 r 1 i 0 j 0 d G B ij d G B ij d λ G B ij, B ij r 1 d G B ij d G B ij d λ G B ij, B ij r 1 λ1 n 0 m j d G v i m j m j 1, by Lemma i 0 j 0 d G v i i0 r 1 λ1 n 0 m j m j m j 1 j0 r 1 λ m n 3 0 n 0 q q j0, by Remark 4 5 Using 1 and the sums A 1, A, A 3 and A 4 in, 3, 4 and 5, respectively, we have, H λ G n 0 q H λ G λ mqn 0 1 λ 1M 1 G 3λ λ r 1 d G v i d G v k n 3 0 qn 0 j 0 Using λ 1 in Theorem 5, we have the following corollary, which is a corrected version of the main theorem proved by Wang and Kang [7] Corollary 6 Let G be a connected graph with n vertices and m edges and let E be the set of edges of G which do not lie on any C 3 of it If n 0 and q are the numbers of vertices and edges of K m0, m 1,, m r 1, r 3, respectively, then DDG K m0, m 1,, m r 1 n 0 q DDG8mqn 0 1 M 1 G 1 d G v i d G v k n 30 qn 0 r 1 j 0 Counterexample: One can easily find the degree distance of the graph K K,, see Fig 3 is 960 For this graph the result given by Wang and Kang [7] is not correct

13 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Figure 3 Tensor product of K and K,, Using Corollary 6, we have the following corollaries Corollary 7 Let G be a connected graph with n vertices and m edges If each edge of G is on a C 3, then DDG K m0, m 1,, m r 1 qn 0 DDG 8mqn 0 1 M 1 G n 30 qn 0 r 1, r 3 j 0 For a triangle free graph, E M 1 G qn 0 r 1 j 0 EG and hence d G v i d G v k Corollary 8 If G is a connected triangle free graph on n vertices and m edges, then DDG K m0, m 1,, m r 1 qn 0 DDG 8mqn 0 1 3M 1G n 3 0, r 3 If m i s, 0 i r 1, in Theorem 5, Corollaries 7 and 8, we have the following corollaries Corollary 9 Let G be a connected graph with n vertices and m edges Let E be the set of edges of G which do not lie on a triangle Then DDG K rs r s 3 r 1DDG 4mrs r s r rs 1 M 1 G 1 d G v i d G v k rs 3 r 1, r 3 Corollary 10 Let G be a connected graph with n vertices and m edges If each edge of G is on a C 3, then DDG K rs r s 3 r 1DDG M 1 Grs 3 r 1 4mrs r s r rs 1, r 3

14 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Corollary 11 If G is a connected triangle free graph on n vertices and m edges, then DDG K rs r s 3 r 1DDG M 1 Grs 3 r 1 4mrs r s r rs 1, r 3 If we consider s 1, in Corollaries 9, 10 and 11, we have the following corollaries Corollary 1 Let G be a connected graph with n vertices and m edges Let E be the set of edges of G which do not lie on a triangle Then DDG K r r r 1DDG M 1 G 1 d G u i d G u k 4mrr 1, r 3 v i v k E Corollary 13 Let G be a connected graph on n vertices with m edges If each edge of G is on a C 3, then DDG K r r r 1DDG M 1 Grr 1 4rr 1 m, where r 3 Corollary 14 If G is a connected triangle free graph on n vertices and m edges, then DDG K r r r 1DDGrr 1M 1 G4rr 1 m, r 3 Using λ 1, in Theorem 5, we obtain the reciprocal degree distance of tensor product of complete multipartite graph K m0, m 1,, m r 1 and a given connected graph G Corollary 15 Let G be a connected graph with n vertices and m edges and let E be the set of edges of G which do not lie on any C 3 of it If n 0 and q are the numbers of vertices and edges of K m0, m 1,, m r 1, r 3, respectively, then RDDG K m0, m 1,, m r 1 n 0 q RDDG mqn 0 1 d G v k n 30 qn 0 r 1 j 0 Using Corollary 15, we have the following corollaries M 1 G 1 1 d G v i Corollary 16 Let G be a connected graph with n vertices and m edges If each edge of G is on a C 3, then RDDG K m0, m 1,, m r 1 qn 0 RDDGmqn 0 1 M 1G n 30 qn 0 r 1, r 3 j 0 Corollary 17 If G is a connected triangle free graph on n vertices and m edges, then RDDG K m0, m 1,, m r 1 qn 0 RDDG mqn 0 1 M 1G n 3 0 qn 0 r 1 j 0, r 3 If m i s, 0 i r 1, in Theorem 5 Corollaries 16 and 17, we have the following corollaries 3

15 K PATTABIRAMAN AND P KANDAN / AUSTRALAS J COMBIN , Corollary 18 Let G be a connected graph with n vertices and m edges Let E be the set of edges of G which do not lie on a triangle Then RDDG K rs r s 3 r 1RDDGmrs r s r rs1 M1 G 1 d 1 G v i v i v k E d G v k rs 3 r 1, r 3 Corollary 19 Let G be a connected graph with n vertices and m edges If each edge of G is on a C 3, then RDDG K rs r s 3 r 1RDDG M 1G rs 3 r 1 mrs r s r rs 1, r 3 Corollary 0 If G is a connected triangle free graph on n vertices and m edges, then RDDG K rs r s 3 r 1RDDG M 1G rs 3 r 1 mrs r s 3 r rs 1, r 3 If we consider s 1, in Corollaries 18, 19 and 0, we have the following corollaries Corollary 1 Let G be a connected graph with n vertices and m edges Let E be the set of edges of G which do not lie on a triangle Then RDDG K r rr 1 rrddg 1M 1G 1 d 1 G u i d G u k r 1m, r 3 v i v k E Corollary Let G be a connected graph on n vertices with m edges If each edge of G is on a C 3, then RDDG K r rr 1 rrddg 1M 1Gr 1m, where r 3 Corollary 3 If G is a connected triangle free graph on n vertices and m edges, then RDDG K r rr 1 rrddg M 3 1G r 1m, r 3 References [1] N Alon and E Lubetzky, Independent set in tensor graph powers, J Graph Theory, , [] Y Alizadeh, A Iranmanesh and T Doslic, Additively weighted Harary index of some composite graphs, Discrete Math, , 6 34 [3] AM Assaf, Modified group divisible designs, Ars Combin , 13 0 [4] B Bresar, W Imrich, S Klavžar and B Zmazek, Hypercubes as direct products, SIAM J Discrete Math , [5] S Chen and W Liu, Extremal modified Schultz index of bicyclic graphs, MATCH Commun Math Comput Chem ,

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