Let G be a connected graph with vertex set V (G). The degree resistance distance of G is defined as DR(G) = ∑ fu,vg V (G)[d(u) +d(v)]R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between u and v. In this paper, we characterize n-vertex unicyclic graphs having minimum and second minimum degree resistance distance.

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

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 exactformula for the wiener index. as application we calculate the schultz index and modified schultz index of this ...

The walk distances in graphs are defined as the result of appropriate transformations of the ∑∞ k=0(tA) k proximity measures, where A is the weighted adjacency matrix of a graph and t is a sufficiently small positive parameter. The walk distances are graphgeodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter t approaches its lim...

Distance on graph is a very useful concept [1]. Both the shortest path distance [2], resistance distance [3] and the mean first-passage distance of a random walker [4-7] can reflect the intrinsic properties of the graph. However, conventional first-passage distance on a graph is based on the basic assumption that the whole network is closed, which means the random walker cannot escape from the ...

the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

Let G be a weighted graph with Laplacian matrix L and signless Laplacian matrix Q. In this note, block representations for the group inverse of L and Q are given. The resistance distance in a graph can be obtained from the block representation of the group inverse of L.