All graphs considered in this paper are finite and undirected, without loops or multiple edges. Let G = (V, E) be a graph. Throughout this paper, let m and n denote the numbers of edges and vertices of graph G, respectively. A connected graph is distance-hereditary if the distance between every two vertices in a connected induced subgraph is the same as that in the original graph. Distance-here...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

Аннотация The Doob graph D(m, n), where m > 0, is the direct product of m copies of The Shrikhande graph and n copies of the complete graph K 4 on 4 vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). In this paper we consider MDS codes in Doob graphs with code distance d ≥ 3. We prove that if 2m + n > 6 and 2 < d < 2m + n, th...

Let G be a connected simple (molecular) graph. The distance d(u, v) between two vertices u and v of G is equal to the length of a shortest path that connects u and v. In this paper we compute some distance based topological indices of H-Phenylenic nanotorus. At first we obtain an exact formula for the Wiener index. As application we calculate the Schultz index and modified Schultz index of this...

Given a graph G with n vertices and k players, each of which is placing a facility on one of the vertices of G, we define the score of the ith player to be the number of vertices for which, among all players, the facility placed by the ith player is the closest. A placement is balanced if all players get roughly the same score. A graph is balanced if all placements on it are balanced. Viewing b...

Let G = (V1, V2, E) be a balanced bipartite graph with 2n vertices. The bipartite binding number of G, denoted by B(G), is defined to be n if G = Kn,n and mini∈ {1,2} min ∅ =S⊆Vi |N(S)| 3/2 and n ≥ 139, then G is bipancyclic; the bound 3...

The diameter of an undirected unweighted graph G = (V,E) is the maximum value of the distance from any vertex u to another vertex v for u, v ∈ V where distance i.e. d(u, v) is the length of the shortest path from u to v in G. DAG, is a directed graph without a cycle. We denote the diameter of an unweighted DAG G = (V,E) by δ(G). The stretch of a DAG G is the length of longest path from u to v i...

We consider the next random process for generating a maximal H-free graph: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving n-vertex graph unless its addition creates a ...

In $1994,$ degree distance  of a graph was introduced by Dobrynin, Kochetova and Gutman. And Gutman proposed the Gutman index of a graph in $1994.$ In this paper, we introduce the concepts of  multiplicative version of degree distance and the multiplicative version of Gutman index of a graph. We find the sharp upper bound for the  multiplicative version of degree distance and multiplicative ver...

The arrangement graph was proposed as a generalization of the star graph topology. In this paper we investigate the topological properties of the (n, k)-arrangement graph &k. It has been shown that the (n, n 2)arrangement graph An,+2 is isomorphic to the n-alternating group graph AGn In addition, the exact value of average distance of A,,& has been derived.