Here we study the normalized Laplacian characteristics polynomial (L-polynomial) for trees and specifically for starlike trees. We describe how the L-polynomial of a tree depends on some topological indices. For which, we also define the higher order general Randić indices for matching and which are different from higher order connectivity indices. Finally we provide the multiplicity of the eig...

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 generalized atom-bond connectivity index of a graph G is denoted by ABCa(G) and defined as the sum of weights ((d(u)+d(v)-2)/d(u)d(v))aa$ over all edges uv∊G. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, we compute sharp bounds for  ABCa index for cacti of order $n$ ...

In this paper, we show that in the class of connected graphs G of order n ≥ 3 having girth at least equal to k, 3 ≤ k ≤ n, the unique graph G having minimum general sum-connectivity index χα(G) consists of Ck and n−k pendant vertices adjacent to a unique vertex of Ck, if−1 ≤ α < 0. This property does not hold for zeroth-order general Randić index Rα(G).

Use vi , i , i , i to denote order, connectivity, edge-connectivity and minimum degree of a graphGi for i=1, 2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are (G1×G2) 1+ 2 and (G1×G2) 1+ 2. This paper improves these results by proving that (G1×G2) min{ 1+ 2, 2+ 1} and (G1×G2)= min{ 1+ 2, 1v2, 2v1} ifG1 andG2 are conne...

Any number that can be uniquely determined by a graph is called invariant. During the last twenty years’ countless mathematical invariants have been characterized and utilized for correlation analysis. However, no reliable examination has embraced to decide, how much these are related with network or molecular graph. In this paper, it will discuss three different variants of bridge networks goo...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...