Let $G$ be a simple graph with vertex set $V=V(G)$ and edge set $E=E(G)$. For a real function $f$ defined on nonnegative real numbers, the vertex-degree function index $H_{f}(G)$ is defined as $$H_{f}(G)=\sum_{u\in V(G)}f(d_{u}).$$ In this paper we introduce the vertex-degree function index $H_{f}(D)$ of a digraph $D$. After giving some examples and basic properties of $H_{f}(D)$, we find the extremal values of $H_{f}$ among all tournaments with a fixed number of vertices, when $f$ is a continuous and convex (or concave) real function on $\left[ 0,+\infty \right)$.

A recently published paper [T. Doslic, this journal 3 (2012) 25-34] considers the Zagreb indices of benzenoid systems, and points out their low discriminativity. We show that analogous results hold for a variety of vertex-degree-based molecular structure descriptors that are being studied in contemporary mathematical chemistry. We also show that these results are straightforwardly obtained by using some identities, well known in the theory of benzenoid hydrocarbons.

The aim of this paper is to first compute some bounds for forgotten index and then to present spectral properties of this topological index. In continuing, we define a new version of energy namely ISI energy corresponding to the ISI index. Finally, we determine some bounds for this new graph invariant.

Let G1 and G2 be simple connected graphs with disjoint vertex sets V(G1) and V(G2 ), respectively. For given vertices a1 ϵ V(G1) and a2 ϵ V(G2 ), a splice of G1 and G2 by vertices a1 and a2 is defined by identifying the vertices a1 and a2 in the union of G1 and G2. In this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

‎Let $G$ be a graph with vertex set $V\left(G \right)$‎. ‎The vertex-degree function index $H_{f}\left(G \right) $ is defined on $G$ as‎: ‎$H_{f}\left(G \right) =\sum_{u\in V\left(G \right)}f\left(d_{u} \right)‎, $‎where $f\left(x \right) $ is a function defined on positive real numbers‎. ‎Our main concern in this paper is to study $H_{f}$ over the set $\mathcal{T}_{n}$ of trees with $n$ vertices‎, ‎over the set $\mathcal{T}_{n, k}$ of trees with $n$ vertices and $k$ branching vertices‎, ‎and over the set $\mathcal{T}^{p}_{n}$ of trees with $n$ vertices and $p$ pendant vertices‎. ‎Namely‎, ‎we will show in each of these sets of trees that it is possible via branching operations to construct a strictly monotone sequence of trees that reaches the extremal values of $H_{f}$‎, ‎when $f\left( x+1\right)-f\left( x\right) $ is a strictly increasing function‎.

ABC index, ABC4 index, Randic connectivity index, Sum connectivity index, GA index, GA5 index, harmonic index, second zagreb index and AZI of Fluorographene are computed.

In the extension of irregularity indices, Abdo et. al. [H. Abdo, S. Brandt, D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014), 201{206] de ned the total irregularity of a graph G = (V; E) as irrt(G) = P   du dv   , where du denotes the vertex degree of a vertex u 2 V (G). In 1 2 u; v2V (G) this paper, we investigate the total irregularity of trees with bounded maximal degree
and state integer linear programming problem which gives standard information about extremal trees and it also calculates the index.

Recently, some techniques such as adding whiskers and attaching graphs to vertices of a given graph, have been proposed for constructing a new vertex decomposable graph. In this paper, we present a new method for constructing vertex decomposable graphs. Then we use this construction to generalize the result due to Cook and Nagel.