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.

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which consecutive edges intersect in a single vertex. We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ1(H) ≥ ( 7 16 + o(1) ) (n 2 ) contains a loose Hamilton cycle. This bound is asym...

The robustness of networks is an important problem that has been studied in a variety of situations, see for instance [3] or the survey [5] and the references therein. A milestone in the study of this problem was the discovery by Albert et al. of the dramatic effect of targeted attacks on networks whose vertex degrees follow a power-law distribution, like the Internet and the World Wide Web [1]...

The eccentricity connectivity index of a molecular graph G is defined as (G) = aV(G) deg(a)ε(a), where ε(a) is defined as the length of a maximal path connecting a to other vertices of G and deg(a) is degree of vertex a. Here, we compute this topological index for some infinite classes of dendrimer graphs.

The atom bond connectivity index of a graph is a new topological index was defined by E. Estrada as ABC(G)  uvE (dG(u) dG(v) 2) / dG(u)dG(v) , where G d ( u ) denotes degree of vertex u. In this paper we present some bounds of this new topological index.