Distance-based indices, including closeness centrality, average path length, eccentricity and average eccentricity, are important tools for network analysis. In these indices, the distance between two vertices is measured by the size of shortest paths between them. However, this measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs (and also directed g...

The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let H be a simple graph. The first Zagreb eccentricity index (E1(H)) is defined to be the summation of squares of the eccentricity of vertices, i.e., E1(H) = ∑u∈V(H) Symmetry 2016, 9, 7; doi: 10.3390/sym9010007 www.mdpi.com/journal/symmetry Article First and Second Zagreb Eccentricity Indices of Thorny Gra...

In 1994, Erd?s, Gyárfás and ?uczak posed the following problem: given disjoint vertex sets V1,…,Vn of size k, with exactly one edge between any pair Vi,Vj, how large can n be such that there will always an independent transversal? They showed maximal is at most (1+o(1))k2, by providing explicit construction these parameters no transversal. also proved a lower bound which smaller 2e-factor. this...

Motivated by a longstanding conjecture of Thomassen, we study how large the average degree graph needs to be imply that it contains C4-free subgraph with at least t. Kühn and Osthus showed an bound which is double exponential in t sufficient. We give short proof this bound, before reducing single exponential. That is, show any G $$2^{ct^{2}\log\ t}$$ (for some constant c > 0) Finally, construct...

if $g$ is a connected graph with vertex set $v$, then the eccentric connectivity index of $g$, $xi^c(g)$, is defined as $sum_{vin v(g)}deg(v)ecc(v)$ where $deg(v)$ is the degree of a vertex $v$ and $ecc(v)$ is its eccentricity. in this paper we show some convergence in probability and an asymptotic normality based on this index in random bucket recursive trees.

The forgotten topological coindex (also called Lanzhou index) is defined for a simple connected graph G as the sum of the terms du2+dv2 over all non-adjacent vertex pairs uv of G, where du denotes the degree of the vertex u in G. In this paper, we present some inequalit...