Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

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...

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ◦H is non-trivial and complete, then G ◦H is edge-transitive if and only if H is the lexicogr...

For graphs G and H, an H-coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H, which connected n-vertex graph with minimum degree δ maximizes the number of H-colorings? We show that for non-regular H and sufficiently large n, the complete bipartite graph Kδ,n−δ is the unique maxim...

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

the tutte polynomial of a graph g, t(g, x,y) is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected. in this paper a simple formula for computing tutte polynomial of a benzenoid chain is presented.

Let Γ be a graph and let G be a group of automorphisms of Γ. The graph Γ is called G-normal if G is normal in the automorphism group of Γ. Let T be a finite non-abelian simple group and let G = T l with l ≥ 1. In this paper we prove that if every connected pentavalent symmetric T -vertex-transitive graph is T -normal, then every connected pentavalent symmetric G-vertex-transitive graph is G-nor...

An edge of a k-connected graph is said to be k-contractible if its contraction yields a k-connected graph. A non-complete k-connected graph possessing no k-contractible edges is called contraction-critical k-connected. Let G be a contraction-critical 7-connected graph with n vertices, and V7 the set of vertices of degree 7 in G. In this paper, we prove that |V7| ≥ n 22 , which improves the resu...

The first reformulated Zagreb index $EM_1(G)$ of a simple graph $G$ is defined as the sum of the terms $(d_u+d_v-2)^2$ over all edges $uv$ of $G .$ In this paper, the various upper and lower bounds for the first reformulated Zagreb index of a connected graph interms of other topological indices are obtained.

The concept of average degree-eccentricity matrix ADE(G) of a connected graph $G$ is introduced. Some coefficients of the characteristic polynomial of ADE(G) are obtained, as well as a bound for the eigenvalues of ADE(G). We also introduce the average degree-eccentricity graph energy and establish bounds for it.