Related papers: The Second Zagreb Indices of Graphs with Given Deg…

Let G be a simple connected molecular graph with vertex set $V(G)$ and edge set $E(G)$. One important modification of classical Zagreb index, called hyper Zagreb index $HM(G)$ is defined as the sum of squares of the degree sum of the…

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is…

This paper presents new lower bounds for the first general Zagreb index $Z_{\alpha}(G)$ involving two, three, and four arbitrary degrees of vertices of a simple graph $G$. For the special cases $\alpha = 2$ and $\alpha = -2$, the results…

The authors Mili$\breve{c}$evi$\acute{c}$ et al. introduced the reformulated Zagreb indices, which is a generalization of classical Zagreb indices of chemical graph theory. In the paper, we characterize the extremal properties of the first…

Let $G = (V, E)$ be a graph. The first Zagreb index and the forgotten topological index of a graph $G$ are defined respectively as $\sum_{u \in V} d^2(u)$ and $\sum_{u \in V} d^3(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. If the…

For a simple graph we introduce notions of the double star sequence, the double star frequently sequence and prove that these sequences are inverses of each other. As a consequence, we express the general second Zagreb index in terms of the…

The second Zagreb index is $M_2(G)=\sum_{uv\in E(G)}d_{G}(u)d_{G}(v)$. It was found to occur in certain approximate expressions of the total $\pi$-electron energy of alternant hydrocarbons and used by various researchers in their QSPR and…

In this paper, we focus on comparing the first and second Zagreb-Fermat eccentricity indices of graphs. We show that $$\frac{\sum_{uv\in E\left( G \right)}{\varepsilon_3\left( u \right) \varepsilon_3\left( v \right)}}{m\left( G \right)}…

In this paper we present a theoretical analysis in order to establish maximal and minimal vectors with respect to the majorization order of particular subsets of \Re ^n: Afterwards we apply these issues to the calcula- tion of bounds for a…

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity…

Let $\prod(G)$ be Multiplicative Zagreb index of a graph G. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which has been the interest of researchers in the filed of material…

Making use of a majorization technique for a suitable class of graphs, we derive upper and lower bounds for some topological indices depending on the degree sequence over all vertices, namely the first general Zagreb index and the first…

For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of…

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is…

The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^\alpha$, and to characterize the set of extremal graphs with respect to them. Our main…

A simple connected graph G is called a p-quasi k-cyclic graph, if there exists a subset S of vertices such that |S|=p, G-S is k-cyclic and there is no a subset S` of V(G) such that |S`|<|S| and G-S` is k-cyclic. The aim of this paper is to…

Applications in chemistry motivated mathematicians to define different topological indices for different types of graphs. The Hyper-Zagreb index (HM) is an important tool as it integrates the first and the second Zagreb indices. In this…

A graph $G$ consists of two parts, the vertices and edges. The vertices constitute the vertex set $V(G)$ and the edges, the edge set. An edge \( e=xy \), \( ev \)-dominates not only the vertices incident to it but also those adjacent to…

In a recent article, Nadeem and Siddique used Chebyshev's sum inequality to establish the Zagreb indices inequality $M_1/n\le M_2/m$ for undirected graphs in the case where the degree sequence $(d_i)$ and the degree-sum sequence $(S_i)$ are…

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two…