The atom-bond connectivity index of graph is a topological index proposed by Estrada et al. as ABC (G)  uvE (G ) (du dv  2) / dudv , where the summation goes over all edges of G, du and dv are the degrees of the terminal vertices u and v of edge uv. In the present paper, some upper bounds for the second type of atom-bond connectivity index are computed.

the atom-bond connectivity index of graph is a topological index proposed by estrada et al.as abc (g)  uve (g ) (du dv  2) / dudv , where the summation goes over all edges ofg, du and dv are the degrees of the terminal vertices u and v of edge uv. in the present paper,some upper bounds for the second type of atom-bond connectivity index are computed.

Albertson [3] has defined the irregularity of a simple undirected graph G = (V,E) as irr(G) = ∑ uv∈E |dG(u)− dG(v)| , where dG(u) denotes the degree of a vertex u ∈ V . Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [13]. For general graphs with n vertices...

We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good approximation of this function using a matrix hyperbolic tangent function. We derive a communicability function, a subgraph centrality and a double-factorial ...