ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.729422 J. Algebra Comb. Discrete Appl. 7(2) • 121–140 Received: 10 August 2017 Accepted: 15 August 2019 Journal of Algebra Combinatorics Discrete Structures and Applications Degree distance and Gutman index of two graph products Research Article Shaban Sedghi, Nabi Shobe Abstract: The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph. 2010 MSC: 05C07, 05C90 Keywords: Degree distance, Adjacency matrix, Distance matrix, Complete product, Strong product 1. Introduction All graphs in this paper are assumed to be undirected, finite and simple. We refer to [2] for graph theoretical notation and terminology not specified here. For a graph G, let V (G), E(G) and G denote the set of vertices, the set of edges and the complement of G, respectively. If G is a connected graph and u,v ∈ V (G), then the distance d(u,v) between u and v is the length of a shortest path connecting u and v. If v is a vertex of a connected graph G, then the eccentricity e(v) of v is defined by e(v) = max{d(u,v) |u ∈ V (G)}. Furthermore, the diameter diam(G) of G is defined by diam(G) = max{e(v) |v ∈ V (G)}. Let G be a finite, simple, connected, undirected graph with p vertices and q edges. In what follows, we say that G is an (p,q)-graph. Let V (G) = {v1,v2, . . . ,vp} and E(G) = {e1,e2, . . . ,eq} be the vertex set and edge set of G, respectively. The adjacency matrix of G is the p×p matrix A = A(G) whose (i,j) entry, denoted by aij, is defined by aij = { 1 if vi and vj are adjacent 0 otherwise. Shaban Sedghi(Corresponding Author); Department of Mathematics, Qaemshahr Branch, Islamic Azad Univer- sity, Qaemshahr, Iran (email: sedghi_gh@yahoo.com, sedghi.gh@qaemiau.ac.ir). Nabi Shobe; Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran (email: nabi_shobe@yahoo.com). 121 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 The distance matrix of G is the p×p matrix DG whose (i,j) entry, denoted by dij, is defined by dij = { dG(vi,vj) if vi 6= vj 0 otherwise, where dG(vi,vj) is the length of a shortest directed path in G from vi to vj. The vertex u is said to be a neighbor of v if they are adjacent. The neighborhood of a vertex v, denoted by NG(v), is the set of all neighbors of v. The degree of a vertex v in a graph G, denoted by dv = dG(v), is the number of vertices in its neighborhood, that is, dG(v) = |N(v)|. The common neighborhood graph con(G) (in short congraph) of a graph G is defined as the graph with V (con(G)) = V (G) and two vertices in con(G) are adjacent if they have a common neighbor in G. For every x,y ∈ V (G), xy ∈ E(con(G)) if and only if NG(x) ∩NG(y) 6= ∅. Some basic properties of congraphs have been established; see [1, 3]. The oldest and most studied degree-based structure descriptors are the first and second Zagreb indices [15], defined as M1(G) = ∑ v∈V (G) (dG(v)) 2 and M2(G) = ∑ uv∈E(G) (dG(u)) (dG(v)) . It has been shown that the first Zagreb index obeys the identity [10] M1(G) = ∑ uv∈E(G) (dG(u) + dG(v)) . The first investigation of the sum of distance between all pairs of vertices of a (connected) graph was done by Harold Wiener in 1947, who realized that there exists a correlation between the boiling points of paraffins and this sum [20]. Eventually, the distance–based graph invariant, W(G) = ∑ {u,v}⊆V (G) d(u,v). For more details, we refer to [8, 11, 13, 19]. The degree distance was introduced by Dobrynin and Kochetova [9] and Gutman [14] as a weighted version of the Wiener index. The degree distance DD(G) of a graph G is defined as DD(G) = ∑ {u,v}⊆V (G) dG(u,v)[dG(u) + dG(v)] = 1 2 ∑ u,v∈V (G) dG(u,v)[dG(u) + dG(v)] with the summation runs over all pairs of vertices of G. The degree distance is also known as the Schultz index in chemical literature; see [21]. In [14], Gutman showed that if G is a tree on n vertices, then DD(G) = 4W(G) −n(n− 1); see [5, 6] and [9]. In [7], Gutman index Gut(G) of a graph G is defined as Gut(G) = ∑ {u,v}⊆V (G) dG(u)dG(v)d(u,v). For more details on Gutman index, we refer to [4, 7, 12]. The relations between the degree distance, Gutman index and Wiener index are shown in the fol- lowing Table 1. The join and strong products are defined as follows. The join or complete product G∨H of two disjoint graphs G and H, is the graph with vertex set V (G) ∪V (H) and edge set E(G) ∪E(H) ∪{uv |u ∈ V (G),v ∈ V (H)}. 122 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 Table 1. Three distance parameters Wiener index W(G) = ∑ {u,v}⊆V (G) dG(u, v) Degree distance DD(G) = ∑ {u,v}⊆V (G) dG(u, v)[dG(u) + dG(v)] Gutman index Gut(G) = ∑ {u,v}⊆V (G) dG(u, v)dG(u)dG(v) The strong product G�H of graphs G and H has the vertex set V (G)×V (H). Two vertices (u,v) and (u′,v′) are adjacent whenever uu′ ∈ E(G) and v = v′, or u = u′ and vv′ ∈ E(H), or uu′ ∈ E(G) and vv′ ∈ E(H). Paulraja and Agnes [16] studied the degree distance of Cartesian and lexicographic products. Later, they [17] investigated the Gutman index of Cartesian and lexicographic products. In this paper, we investigate the degree distance and Gutman index of strong and complete product graphs. 2. Preliminary We define, N1(G) = ∑ v∈V (G) dG(v) dcon(G)(v) and N2(G) = ∑ uv∈E(con(G)) dG(u) dG(v). Definition 2.1. Let A = [aij]m×n. Then, we define S(A) = ∑ 1≤i≤m, 1≤j≤n aij. The following lemma is immediate. Lemma 2.2. Let A = [aij]n×n and B = [bij]n×n. Then (1) S(AT ) = S(A) and S(αA) = αS(A) for every α ∈ R; (2) S(A + B) = S(A) + S(B). Lemma 2.3. Let G be a (p,q)-graph, and let con(G) be a (p,q′)-graph. Let A,B,K be the adjacency matrices of G,con(G),Kp, respectively. Then (1) S(A) = 2q; (2) S(A2) = M1(G); (3) S(AB) = N1(G); (4) S(A3) = 2M2(G); (5) S(AK) = 2q(p− 1). (6) S(AB A) = 2N2(G). Proof. For (1), we have S(A) = ∑ 1≤i,j≤p aij = p∑ i=1 p∑ j=1 aij = p∑ i=1 dvi = 2q. 123 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 For (2), we have S(A2) = ∑ 1≤i,j≤p a (2) ij = ∑ 1≤i,j≤p p∑ k=1 aik akj = p∑ k=1 p∑ i=1 aik p∑ j=1 akj = p∑ k=1 dvk dvk = M1(G). For (3), we have S(AB) = ∑ 1≤i,j≤p p∑ k=1 aik bkj = p∑ k=1 p∑ i=1 aik p∑ j=1 bkj = p∑ k=1 dvk dconGvk = N1(G). For (4), we have M2(G) = ∑ vivj∈E(G) dvi dvj = 1 2 ∑ 1≤i,j≤p dvi dvj aij = 1 2 p∑ i=1 p∑ j=1 ( p∑ k=1 aki )( p∑ s=1 ajs ) aij = 1 2 p∑ k=1 p∑ j=1 p∑ s=1 ajs p∑ i=1 akiaij. Since ∑p i=1 akiaij is the entry tkj of matrix A 2, it follows that M2(G) = 1 2 p∑ k=1 p∑ j=1 p∑ s=1 ajs p∑ i=1 akiaij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 tkjajs = 1 2 p∑ k=1 p∑ s=1 a (3) ks = 1 2 S(A3). For (5), we have S(AK) = ∑ 1≤i,j≤p p∑ r=1 air krj = p∑ r=1 p∑ i=1 air p∑ j=1 krj = p∑ r=1 dvr (p− 1) = 2q(p− 1). For (6), we have N2(G) = ∑ vivj∈E(conG) dvi dvj = 1 2 ∑ 1≤i,j≤p dvi dvj bij = 1 2 p∑ i=1 p∑ j=1 ( p∑ k=1 aki )( p∑ s=1 asj ) bij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 asj p∑ i=1 akibij. 124 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 Since, ∑p i=1 akibij is the entry tkj of matrix AB, hence N2(G) = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 asj p∑ i=1 akibij = 1 2 p∑ k=1 p∑ s=1 p∑ j=1 tkjajs = 1 2 p∑ k=1 p∑ s=1 fks = 1 2 S(AB A), where ∑p i=1 tkjajs is the entry fks of matrix AB A. The following result for classical distance are from the book [18]. Lemma 2.4. [18] Let (u,v) and (u′,v′) be two vertices of G1 � G2. Then dG1�G2 ((u,v), (u ′,v′)) = max{dG1 (u,u ′),dG2 (v,v ′)}. For G2 = Kp, the following result is immediate. Corollary 2.5. Let Kp be a complete graph, and let (u,v) and (u′,v′) be two vertices of G � Kp. Then dG�Kp ((u,v), (u ′,v′)) =   dG(u,u ′) if u 6= u′, 1 if u = u′ and v 6= v′, 0 if u = u′ and v = v′. 3. Main results In this section, we give our main results and their proofs. 3.1. Relation between degree distance and Gutman index We first define a matrix, which will be used later. Definition 3.1. Let G(V,E) be a graph with order n and m edges. For k = 1, 2, · · · ,α where α denotes the diameter of graph G, we define Ak = [a k ij]n×n, where akij = { 1 d(vi,vj) = k 0 otherwise. The following results are easily seen. Observation 3.1. Let A and DG be the adjacency matrix and the distance matrix of a graph G, respec- tively. Then (1) A1 = A; (2) DG = A1 + 2A2 + · · · + αAα; (3) A1 + A2 + · · · + Aα = K, where K is the adjacency matrix complete graph Kn; (4) if diam(G) = 2 then DG = A + 2A. 125 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 Lemma 3.2. Let G be a graph containing no triangles, and let A,B,K be the adjacency matrix of G,con(G),Kn, respectively. Then (1) for every u,v ∈ V (G), dG(u,v) = 2 if and only if uv ∈ E(con(G)); (2) if diam(G) = 3 then DG = 3K − 2A−B. Proof. (1) Suppose dG(u,v) = 2. Then there exists a vertex x ∈ V (G) such that x /∈ {u,v} and ux,xv ∈ E(G), and hence N(u)∩N(v) 6= ∅. Therefore, we have uv ∈ E(con(G)). Conversely, we suppose uv ∈ E(con(G)). Then N(u) ∩N(v) 6= ∅, and hence there exists a vertex x ∈ N(u) ∩N(v). Note that ux,xv ∈ E(G). Therefore, d(u,v) ≤ 2. If d(u,v) = 1, then we have a triangle, a contradiction. So dG(u,v) = 2, as desired. (2) From Observation 3.1, we have DG = A1 + 2A2 + 3A3. Note that A1 = A, A2 = B and A + B + A3 = K. Therefore, DG = 3K − 2A−B. Lemma 3.3. Let G(p,q) be a graph, and let A,DG be the adjacency matrix and the distance matrix of a graph G, respectively. Then (1) S(ADG) = DD(G); (2) if diam(G) = 2 then DD(G) = 4(p− 1)q −M1(G); (3) if diam(G) = 3 and G has no triangles, then DD(G) = 6q(p− 1) − 2M1(G) −N1(G). Proof. (1) Since S(ADG) = p∑ i=1 p∑ j=1 p∑ k=1 aikdkj = ∑ 1≤j,k≤p p∑ i=1 aik d(vk,vj) = ∑ 1≤j,k≤p d(vk) d(vk,vj) and S(DG A) = p∑ i=1 p∑ j=1 p∑ k=1 dikakj = ∑ 1≤i,k≤p d(vi,vk) p∑ j=1 akj = ∑ 1≤i,k≤p d(vi,vk) d(vk) = ∑ 1≤j,k≤p d(vk,vj) d(vj), it follows that 2S(ADG) = S(ADG) + S((ADG) T ) = S(ADG) + S(DG A) = ∑ 1≤j,k≤p d(vk,vj)[d(vk) + d(vj)] = 2 ∑ {vk,vj}⊆V (G) d(vk,vj)[d(vk) + d(vj)] = 2DD(G). For (2), we have DD(G) = S(ADG) = S(A(A + 2A)) = 2S(A(A + A)) −S(A2) = 2S(AK) −S(A2) = 4(p− 1)q −M1(G). 126 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 For (3), we have DD(G) = S(ADG) = S(A(3K − 2A−B)) = 3S(AK) − 2S(A2) −S(AB) = 6q(p− 1) − 2M1(G) −N1(G). Lemma 3.4. Let G be a (p,q)-graph and A be the adjacency matrix of G. Then Gut(G) = 1 2 S(ADG A) Proof. S(ADG A) = p∑ i=1 p∑ j=1 p∑ k=1 aik p∑ s=1 dksasj = p∑ k=1 p∑ s=1 p∑ i=1 aik p∑ j=1 asjdks = p∑ k=1 p∑ s=1 dG(vk) ·dG(vs) ·d(vk,vs) = 2 ∑ {vk,vs}⊆V dG(vk)dG(vs)d(vk,vs) = 2Gut(G). Corollary 3.5. Let G(p,q) be a graph, then δ 2 ≤ Gut(G) DD(G) ≤ ∆ 2 . Proof. Since, S(ADG) = DD(G) and Gut(G) = 12S(ADG A), hence 2Gut(G) −DD(G) = S(ADG A) −S(ADG) = S(ADG(A− I)) = ∑ 1≤i,j≤p p∑ k=1 tik(akj − 1kj) = ∑ 1≤i,k≤p tik(dG(vk) − 1). Therefore, (δ − 1) ∑ 1≤i,k≤p tik ≤ 2Gut(G) −DD(G) ≤ (∆ − 1) ∑ 1≤i,k≤p tik. Hence, (δ − 1)S(ADG) ≤ 2Gut(G) −DD(G) ≤ (∆ − 1)S(ADG). Thus, δDD(G) ≤ 2Gut(G) ≤ ∆DD(G), that is δ 2 ≤ Gut(G) DD(G) ≤ ∆ 2 . 127 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 3.2. For degree distance In this subsection, we study the degree distance of strong product graphs. We first begin with an easy case. Theorem 3.6. Let G be a connected graph with p1 vertices and q1 edges, and Kp be a complete graph with order p. Then DD(G � Kp) = p 3DD(G) + 2p2(p− 1)[W(G) + q1] + p1p(p− 1)2. Proof. Let V (G) = V1 and V (Kp) = V2. From the definition of strong product and Corollary 2.5, we have DD(G � Kp) = ∑ {(a,b),(c,d)}⊆V1×V2 [dG�Kp (a,b) + dG�Kp (c,d)]dG�Kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG(a) + dKp (b) + dG(a)dKp (b) + dG(c) + dKp (d) + dG(c)dKp (d)] = ∑ {(a,b),(c,d)}⊆V1×V2,a 6=c [dG(a) + p− 1 + dG(a)(p− 1) + dG(c) + p− 1 + dG(c)(p− 1)] ·dG(a,c) + ∑ {(a,b),(c,d)}⊆V1×V2,a=c [dG(a) + p− 1 + dG(a)(p− 1) + dG(c) + p− 1 + dG(c)(p− 1)] · 1 = ∑ {(a,b),(c,d)}⊆V1×V2,a 6=c [p(dG(a) + dG(c)) + 2(p− 1)]dG(a,c) + ∑ {(a,b),(c,d)}⊆V1×V2,a=c [2pdG(a) + 2(p− 1)] = p ∑ {(a,b),(c,d)}⊆V1×V2,a 6=c [dG(a) + dG(c)]dG(a,c) + 2(p− 1) ∑ {(a,b),(c,d)}⊆V1×V2,a6=c dG(a,c) +2p ∑ {(a,b),(c,d)}⊆V1×V2,a=c dG(a) + 2(p− 1) ∑ {(a,b),(c,d)}⊆V1×V2,a=c 1 = p3DD(G) + 2p2(p− 1)W(G) + 2p · p(p− 1) 2 · 2q1 + 2p1(p− 1) · p(p− 1) 2 = p3DD(G) + 2p2(p− 1)[W(G) + q1] + p1p(p− 1)2. For the strong product of two general graphs, we have the following. Theorem 3.7. Let G1 be a connected graph with p1 vertices and q1 edges, and G2 be a connected graph 128 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 with p2 vertices and q2 edges. Then max { DD(G1)[2p2q2 + p 2 2] + 4p2q2W(G1) + 2p2(p2 − 1)q1W(G2) + DD(G2)(2q1 + p1), DD(G2)[2p1q1 + p 2 1] + 4p1q1W(G2) + 2p1(p1 − 1)q2W(G1) + DD(G1)(2q2 + p2) } ≤ DD(G1 � G2) ≤ (2q2 + p2)(p2 + 1)DD(G1) + q2(4p2 + 2p21 − 2p1)W(G1) +(2q1 + p1)(p1 + 1)DD(G2) + q1(4p1 + 2p 2 2 − 2p2)W(G2). Moreover, the lower bound is sharp. In particular, if G be a connected graph with p vertices and q edges, then (2q + p)(p + 1)DD(G) + 2pq(p + 1)W(G) ≤ DD(G � G) ≤ 2 { (2q + p)(p + 1)DD(G) + 2pq(p + 1)W(G) } . Proof. From Lemma 2.4 and the definition of degree distance, we have DD(G1 � G2) = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1�G2 (a,b) + dG1�G2 (c,d)]dG1�G2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)] · max{dG1 (a,c),dG2 (b,d)} ≥ max { ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d), ∑ {(a,b),(c,d)}⊆V1×V2 b6=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 b=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) } 129 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 = max { ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG1 (a) + dG1 (c)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG2 (b) + dG2 (d)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(a,d)}⊆V1×V2 [2dG1 (a) + (dG1 (a) + 1)(dG2 (b) + dG2 (d))]dG2 (b,d), ∑ {(a,b),(c,d)}⊆V1×V2 b6=d [dG1 (a) + dG1 (c)]dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 b 6=d [dG2 (b) + dG2 (d)]dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 b6=d [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG2 (b,d) + ∑ {(a,b),(c,b)}⊆V1×V2 [2dG2 (b) + (dG2 (b) + 1)(dG1 (a) + dG1 (c))]dG1 (a,c) } = max { p22DD(G1) + 4p2q2W(G1) + ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG1 (a,c) +2p2(p2 − 1)q1W(G2) + DD(G2)(2q1 + p1), p21DD(G2) + 4p1q1W(G2) + ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG2 (b,d) +2p1(p1 − 1)q2W(G1) + DD(G1)(2q2 + p2) } Since ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG1 (a,c) = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a)dG2 (b) ·dG1 (a,c)] + ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (c)dG2 (d) ·dG1 (a,c)] 130 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 = ∑ {a,c}⊆V1 dG1 (a)dG1 (a,c) · ∑ {b,d}⊆V2 dG2 (b) + ∑ {a,c}⊆V1 dG1 (c)dG1 (a,c) · ∑ {b,d}⊆V2 dG2 (d) = 2p2q2 · ∑ {a,c}⊆V1 dG1 (a)dG1 (a,c) + 2p2q2 · ∑ {a,c}⊆V1 dG1 (c)dG1 (a,c) = 2p2q2 ( ∑ {a,c}⊆V1 [dG1 (a) + dG1 (c)]dG1 (a,c) ) = 2p2q2 ·DD(G1) and similarly ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a)dG2 (b) + dG1 (c)dG2 (d)]dG2 (b,d) = 2p1q1 ( ∑ {b,d}⊆V2 [dG2 (b) + dG2 (d)]dG2 (b,d) ) = 2p1q1DD(G2), it follows that DD(G1 � G2) ≥ max { DD(G1)[2p2q2 + p 2 2] + 4p2q2W(G1) + 2p2(p2 − 1)q1W(G2) + DD(G2)(2q1 + p1), DD(G2)[2p1q1 + p 2 1] + 4p1q1W(G2) + 2p1(p1 − 1)q2W(G1) + DD(G1)(2q2 + p2) } . Also, we have DD(G1 � G2) = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1�G2 (a,b) + dG1�G2 (c,d)]dG1�G2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)] · max{dG1 (a,c),dG2 (b,d)} ≤ ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b) + dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d) = (2q2 + p2)(p2 + 1)DD(G1) + q2(4p2 + 2p 2 1 − 2p1)W(G1) +(2q1 + p1)(p1 + 1)DD(G2) + q1(4p1 + 2p 2 2 − 2p2)W(G2). To show the sharpness of the lower bounds of Theorem 3.7, we consider the following example. Example 1. Let G be a complete graph of order n. If n = 2, then G = K2 and G � G = K4, and hence DD(G � G) = 36 = (2q + p)(p + 1)DD(G) + 2pq(p + 1)W(G). If n = 3, then G = K3 and G � G = K9, and hence DD(G � G) = 576 = (2q + p)(p + 1)DD(G) + 2pq(p + 1)W(G). From the proof of Theorem 3.7, one can check that Kn � Kn is an sharp example of the lower bound. 131 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 3.3. For Gutman index In this subsection, we study the Gutman index of strong product graphs. We first begin with an easy case. Theorem 3.8. Let G be a connected graph with p1 vertices and q1 edges, and Kp be a complete graph with p vertices. Then Gut(G � Kp) = p 4Gut(G) + p3(p− 1)DD(G) + p2(p− 1)2 ·W(G) + p3(p− 1) 2 M1(G) + p(p− 1)3 2 p1 + 2p 2(p− 1)2q1. Proof. Let V (G) = V1 and V (Kp) = V2. From the definition of strong product and Corollary 2.5, we have Gut(G � Kp) = ∑ {(a,b),(c,d)}⊆V1×V2 dG�Kp (a,b) ·dG�Kp (c,d) ·dG�Kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG(a) + dKp (b) + dG(a)dKp (b)] · [dG(c) + dKp (d) + dG(c)dKp (d)] ·dG�Kp [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2,a 6=c [pdG(a) + p− 1] · [pdG(c) + p− 1] ·dG(a,c) + ∑ {(a,b),(c,d)}⊆V1×V2,a=c [pdG(a) + p− 1] · [pdG(a) + (p− 1)] · 1 = p2 ∑ {(a,b),(c,d)}⊆V1×V2,a6=c dG(a)dG(c)dG(a,c) +p(p− 1) ∑ {(a,b),(c,d)}⊆V1×V2,a 6=c [dG(a) + dG(c)]dG(a,c) +(p− 1)2 ∑ {(a,b),(c,d)}⊆V1×V2,a6=c dG(a,c) + p 2 ∑ {(a,b),(c,d)}⊆V1×V2,a=c d2G(a) + ∑ {(a,b),(c,d)}⊆V1×V2,a=c (p− 1)2 + 2p(p− 1) ∑ {(a,b),(c,d)}⊆V1×V2,a=c dG(a) = p4Gut(G) + p3(p− 1)DD(G) + p2(p− 1)2 ·W(G) + p3(p− 1) 2 M1(G) + p(p− 1)3 2 p1 + 2p 2(p− 1)2q1. For the strong product of two general graphs, we have the following. 132 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 Theorem 3.9. Let G1 be a connected graph with p1 vertices and q1 edges, and G2 be a connected graph with p2 vertices and q1 edges. Then max { Gut(G1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)DD(G1) + 4q 2 2W(G1) + M1(G1)W(G2) + [2q1 + M1(G1)]DD(G2) + [p1 + 4q1 + M1(G1)]Gut(G2), Gut(G2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)DD(G2) + 4q 2 1W(G2) +M2(G2)W(G1) + [2q2 + M2(G2)]DD(G1) + [p2 + 4q2 + M2(G2)]Gut(G1) } ≤ Gut(G1 � G2) ≤ Gut(G1)[p22 + 4p2q2 + 4q 2 2 + p2 + 4q2 + M2(G2)] + [2p2q2 + 4q 2 2 + 2q2 + M2(G2)]DD(G1) +[4q22 + M2(G2)]W(G1) + Gut(G2)[p 2 1 + 4p1q1 + 4q 2 1 + p1 + 4q1 + M1(G1)] +[2p1q1 + 4q 2 1 + 2q1 + M1(G1)]DD(G2) + [4q 2 1 + M1(G1)]W(G2). In particular, if G be a connected graph with p vertices and q edges, then Gut(G)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + M1(G)]DD(G) + [4q 2 + M1(G)]W(G) ≤ Gut(G � G) ≤ 2 { Gut(G)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + M1(G)]DD(G) + [4q 2 + M1(G)]W(G) } Proof. Let V (G1) = V1 and V (G2) = V2. From the definition of strong product and Lemma 2.4, we have Gut(G1 � G2) = ∑ {(a,b),(c,d)}⊆V1×V2 dG1�G2 (a,b) ·dG1�G2 (c,d) ·dG1�G2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)] · max{dG1 (a,c),dG2 (b,d)} ≥ max { ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d), ∑ {(a,b),(c,d)}⊆V1×V2 b 6=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 b=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) } 133 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 = max { ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(a,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (a) + dG2 (d) + dG1 (a)dG2 (d)]dG2 (b,d), ∑ {(a,b),(c,d)}⊆V1×V2 b6=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d) + ∑ {(a,b),(c,b)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (b) + dG1 (c)dG2 (b)]dG1 (a,c) } = max{X1 + X2,Y1 + Y2}, where X1 = ∑ {(a,b),(c,d)}⊆V1×V2 a6=c [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c), X2 = ∑ {(a,b),(a,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (a) + dG2 (d) + dG1 (a)dG2 (d)]dG2 (b,d), Y1 = ∑ {(a,b),(c,d)}⊆V1×V2 b6=d [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d), and Y2 = ∑ {(a,b),(c,b)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (b) + dG1 (c)dG2 (b)]dG1 (a,c). Note that X1 = ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG1 (a) ·dG1 (c) ·dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a 6=c dG1 (a) ·dG2 (d) ·dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG1 (a)dG1 (c)dG2 (d)dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG2 (b)dG1 (c)dG1 (a,c) 134 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG2 (b)dG2 (d)dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG2 (b)dG1 (c)dG2 (d)dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG1 (a)dG2 (b)dG1 (c)dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG1 (a)dG2 (b)dG2 (d)dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 a6=c dG1 (a)dG2 (b)dG1 (c)dG2 (d)dG1 (a,c) = p22Gut(G1) + 4p2q2Gut(G1) + 4q 2 2Gut(G1) + 2p2q2DD(G1) + 4q 2 2DD(G1) + 4q 2 2W(G1) = Gut(G1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)DD(G1) + 4q 2 2W(G1) and X2 = ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a) 2 ·dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a) ·dG2 (d) ·dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a) 2dG2 (d)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG2 (b)dG1 (a)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG2 (b)dG2 (d)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG2 (b)dG1 (a)dG2 (d)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a) 2dG2 (b)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a)dG2 (b)dG2 (d)dG2 (b,d) + ∑ {(a,b),(c,d)}⊆V1×V2 dG1 (a) 2dG2 (b)dG2 (d)dG2 (b,d) = M1(G1)W(G2) + 2q1DD(G2) + M1(G1)DD(G2) + p1Gut(G2) + 4q1Gut(G2) +M1(G1)Gut(G2) = M1(G1)W(G2) + [2q1 + M1(G1)]DD(G2) + [p1 + 4q1 + M1(G1)]Gut(G2). Similarly, we have Y1 = Gut(G2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)DD(G2) + 4q 2 1W(G2) and Y2 = M2(G2)W(G1) + [2q2 + M2(G2)]DD(G1) + [p2 + 4q2 + M2(G2)]Gut(G1). Then Gut(G1 � G2) ≥ max { Gut(G1)(p 2 2 + 4p2q2 + 4q 2 2) + (2p2q2 + 4q 2 2)DD(G1) + 4q 2 2W(G1) +M1(G1)W(G2) + [2q1 + M1(G1)]DD(G2) + [p1 + 4q1 + M1(G1)]Gut(G2), Gut(G2)(p 2 1 + 4p1q1 + 4q 2 1) + (2p1q1 + 4q 2 1)DD(G2) + 4q 2 1W(G2) +M2(G2)W(G1) + [2q2 + M2(G2)]DD(G1) + [p2 + 4q2 + M2(G2)]Gut(G1) } 135 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 and Gut(G1 � G2) = ∑ {(a,b),(c,d)}⊆V1×V2 dG1�G2 (a,b) ·dG1�G2 (c,d) ·dG1�G2 [(a,b), (c,d)] = ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)] · max{dG1 (a,c),dG2 (b,d)} ≤ ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG1 (a,c) + ∑ {(a,b),(c,d)}⊆V1×V2 [dG1 (a) + dG2 (b) + dG1 (a)dG2 (b)] · [dG1 (c) + dG2 (d) + dG1 (c)dG2 (d)]dG2 (b,d) = X1 + X2 + Y1 + Y2 ≤ Gut(G1)[p22 + 4p2q2 + 4q 2 2 + p2 + 4q2 + M2(G2)] + [2p2q2 + 4q 2 2 + 2q2 + M2(G2)]DD(G1) +[4q22 + M2(G2)]W(G1) + Gut(G2)[p 2 1 + 4p1q1 + 4q 2 1 + p1 + 4q1 + M1(G1)] +[2p1q1 + 4q 2 1 + 2q1 + M1(G1)]DD(G2) + [4q 2 1 + M1(G1)]W(G2). To show the sharpness of the lower bounds of Theorem 3.9, we consider the following example. Example 1. Let G be a complete graph of order n. If n = 2, then G = K2 and G � G = K4, and hence Gut(G�G) = 54 = Gut(G)[p2 +4pq+4q2 +p+4q]+[2pq+4q2 +2q+M1(G)]DD(G)+[4q 2 +M1(G)]W(G). If n = 3, then G = K3 and G � G = K9, and hence Gut(G � G) = 2304 = Gut(G)[p2 + 4pq + 4q2 + p + 4q] + [2pq + 4q2 + 2q + M1(G)]DD(G) + [4q 2 + M1(G)]W(G). From the proof of Theorem 3.9, one can check that Kn � Kn is an sharp example of the lower bound. 3.4. For complete product We first give the following lemma. Lemma 3.10. (1) If A = [aij]n×m be any matrix and I = [1]p×n, then S(I A) = pS(A); (2) If A = [aij]m×n and I = [1]n×p, then S(AI) = pS(A); (3) If A = [aij]p×m, I = [1]m×n and B = [bij]n×q, then S(AI B) = S(A) · S(B). In particular, if A = [aij]n×n then S(AI A) = S(A)2. Proof. For (1), we have S(I A) = p∑ i=1 m∑ j=1 n∑ k=1 1ikakj = p∑ i=1 n∑ k=1 m∑ j=1 akj = p∑ i=1 S(A) = pS(A). 136 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 For (2), we have S(AI) = m∑ i=1 p∑ j=1 n∑ k=1 aik1kj = p∑ j=1 m∑ i=1 n∑ k=1 aik = p∑ j=1 S(A) = pS(A). For (3), we have S(AI B) = p∑ i=1 q∑ j=1 m∑ k=1 aik n∑ s=1 1ksbsj = p∑ i=1 m∑ k=1 aik q∑ j=1 n∑ s=1 bsj = S(A) ·S(B). Corollary 3.11. Let G be a (p,q)-graph and A and K be the adjacency matrix of G and Kp respectively. Let I = [1]p×p and Ip be the identity matrix. Then S(AKA) = 4q2 −M1(G). Proof. By Lemma 3.10, we have S(AK A) = S(A(I − Ip)A) = S(AI A) −S(A2) = S(A) 2 −S(A2) = 4q2 −M1(G). Theorem 3.12. Let G be a (p,q)-graph, then (1) If diam(G) = 2, then Gut(G) = 4q2 −M1(G) −M2(G). (2) If diam(G) = 3 and G has no cycles of size 3 then Gut(G) = 6q2 − 3 2 M1(G) − 2M2(G) −N2(G). Proof. (1) By Lemma 3.4 and Observation 3.1, we have: 2Gut(G) = S(ADG A) = S(A(A + 2A)A) = S(A 3) + 2S(AAA) = 2S(A(A + A)A) −S(A3) = 2S(AK A) −S(A3) = 8q2 − 2M1(G) − 2M2(G). (2) By Lemma 3.4 and Observation 3.1, we have: 2Gut(G) = S(ADG A) = S(A(3K − 2A−B)A) = 3S(AK A) − 2S(A3) −S(AB A) = 12q2 − 3M1(G) − 4M2(G) − 2N2(G). 137 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 Remark 3.13. Let A1 = [aij]n1×n1 and A2 = [bij]n2×n2 be the adjacency matrix of G1 and G2, respec- tively. Let DG be distance matrix of graph G = G1 ∨G2. Let I1 = [1]n1×n1, I2 = [1]n2×n2, I ′ 1 = [1]n1×n2, I ′ 2 = [1]n2×n1 and In be identity matrices. Then DG = ( 2I1 −A1 − 2In1, I ′ 1 I ′ 2, 2I2 −A2 − 2In2 ) is distance matrix of G1 ∨G2. Theorem 3.14. Let G1 be a graph with order n1 and m1 edges and G2 be a graph with order n2 and m2 edges. Then DD(G1 ∨G2) = 4(n1 + n2 − 1)(m1 + m2 + n1n2) −M(G1 ∨G2). Proof. Since diam(G1 ∨G2) = 2, it follows from Lemma 3.3 that DD(G1 ∨G2) = 4(n1 + n2 − 1)(m1 + m2 + n1n2) −M1(G1 ∨G2). For computing M1(G1 ∨G2), let A be the adjacency matrix of graph G = G1 ∨G2. Then M1(G1 ∨G2) = S(A2) = S    A1, I′1 I ′ 2, A2    A1, I′1 I ′ 2, A2     = S   A12 + I′1I′2, A1I′1 + I′1A2 I ′ 2A1 + A2I ′ 2, I ′ 2I ′ 1 + A2 2   = S(A1 2) + S(I ′ 1I ′ 2) + S(A1I ′ 1) + S(I ′ 1A2) + S(I ′ 2A1) + S(A2I ′ 2) + S(I ′ 2I ′ 1) + S(A2 2) = M1(G1) + n 2 1n2 + 4n2m1 + 4n1m2 + n 2 2n1 + M1(G2). Theorem 3.15. Let G1 be an (n1,m1)-graph and let G2 be an (n2,m2)-graph. Then Gut(G1 ∨G2) = 4(m1 + m2 + n1n2)2 −M1(G1 ∨G2) −M2(G1 ∨G2). Proof. Let A1 = [aij]n1×n1 and A2 = [bij]n2×n2 be the adjacency matrix of G1 and of G2 respectively. Let DG be distance matrix of graph G = G1 ∨G2. If we set I1 = [1]n1×n1, I2 = [1]n2×n2, I ′ 1 = [1]n1×n2, I ′ 2 = [1]n2×n1 and In be identity matrix, then it follows from Theorem 3.12 that Gut(G1 ∨G2) = 4(m1 + m2 + n1n2)2 −M1(G1 ∨G2) −M2(G1 ∨G2), since diam(G1 ∨G2) = 2. For computing M2(G1 ∨G2), let A be the adjacency matrix of graph G = G1 ∨G2. Then 138 S. Sedghi, N. Shobe / J. Algebra Comb. Discrete Appl. 7(2) (2020) 121–140 2M2(G1 ∨G2) = S(A3) = S    A1, I′1 I ′ 2, A2    A1, I′1 I ′ 2, A2    A1, I′1 I ′ 2, A2     = S     A12 + I′1I′2 A1I′1 + I′1A2 I ′ 2A1 + A2I ′ 2 I ′ 2I ′ 1 + A2 2    A1, I′1 I ′ 2, A2     = S     A13 + I′1I′2A1 + A1I′1I′2 + I′1A2I′2 A12I′1 + I′1I′2I′1 + A1I′1A2 + I′1A22 I ′ 2A1 2 + A2I ′ 2A1 + I ′ 2I ′ 1I ′ 2 + A2 2I ′ 2 I ′ 2A1I ′ 1 + A2I ′ 2I ′ 1 + I ′ 2I ′ 1A2 + A2 3     = S(A1 3) + S(I ′ 1I ′ 2A1) + S(A1I ′ 1I ′ 2) + S(I ′ 1A2I ′ 2) + S(A1 2I ′ 1) + S(I ′ 1I ′ 2I ′ 1) + S(A1I ′ 1A2) + S(I ′ 1A2 2) + S(I ′ 2A1 2) + S(A2I ′ 2A1) + S(I ′ 2I ′ 1I ′ 2) + S(A2 2I ′ 2) + S(I ′ 2A1I ′ 1) + S(A2I ′ 2I ′ 1) + S(I ′ 2I ′ 1A2) + S(A2 3). From Lemma 3.10, we have 2M2(G1 ∨G2) = 2M2(G1) + 4n1n2m1 + 2n2M1(G1) + 2n21n 2 2 + 8m1m2 + 2n21m2 + 2n1M1(G2) + 2n 2 2m1 + 4n1n2m2 + 2M2(G2), and hence M2(G1 ∨G2) = M2(G1) + M2(G2) + n2M1(G1) + n1M1(G2) + 2n1n2m2 + 2n1n2m1 + n 2 1n 2 2 + 4m1m2 + n 2 1m2 + n 2 2m1 = M2(G1) + M2(G2) + n2M1(G1) + n1M1(G2) + (n1n2 + 2m2)(n1n2 + 2m1) + n 2 1m2 + n 2 2m1. References [1] A. Alwardi, B. Arsić, I. Gutman, N. D. Soner, The common neighborhood graph and its energy, Iran. J. Math. Sci. Inf. 7 (2012) 1–8. [2] J. A. Bondy, U. S. R. Murty, Graph theory, Springer, New York, 2008. [3] A. S. Bonifácio, R. R. Rosa, I. Gutman, N. M. M. de Abreu, Complete common neighborhood graphs, Proceedings of Congreso Latino-Iberoamericano de Investigaci on Operativa and Simposio Brasileiro de Pesquisa Operacional (2012) 4026–4032. [4] S. Chen, Cacti with the smallest, second smallest, and third smallest Gutman index, J. Combin. 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