The k-th semi-total point graph R(G) of a graph G, is a graph obtained from G by adding k vertices corresponding to each edge and connecting them to the endpoints of the edge considered. In this paper, we obtain formulas for the resistance distance and Kirchhoff index of R(G).

let g be a graph. the first zagreb m1(g) of graph g is defined as: m1(g) = uv(g) deg(u)2. in this paper, we prove that each even number except 4 and 8 is a first zagreb index of a caterpillar. also, we show that the fist zagreb index cannot be an odd number. moreover, we obtain the fist zagreb index of some graph operations.

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

In this paper, we prove the existence of fixed point for Chatterjea type mappings under $c$-distance in cone metric spaces endowed with a graph. The main results extend, generalized and unified some fixed point theorems on $c$-distance in metric and cone metric spaces.

A new family of distances for graph vertices is proposed. These distances reduce to the shortest path distance and to the resistance distance at the extreme values of the family parameter. The most important property of them is that they are graphgeodetic: d(i, j)+d(j, k) = d(i, k) if and only if every path from i to k passes through j. The construction of the distances is based on the matrix f...