Related papers: Upper and lower bounds for topological indices on …
The revised Szeged index of a graph $G$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex…
The concepts of geometric-arithmetic and harmonic indices were introduced in the area of chemical graph theory recently. They have proven to correlate well with physical and chemical properties of some molecules. The aim of this paper is to…
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…
The edge Szeged index of a graph $G$ is defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the…
Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\not\in G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this article,…
Let G be a graph with vertex set V (G) and edge set E(G). The first generalized multiplicative Zagreb index of G is M_1(G) and the second multiplicative Zagreb index is M_2(G). The multiplicative Zagreb indices have been the focus of…
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…
Topological relations between three degree-based invariants of a connected graph G are investigated. We present novel inequalities including M1(G), M2(G) and F(G), and show that in all cases equality holds if G is a regular or a semiregular…
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…
Albertson has defined the irregularity of a simple undirected graph $G=(V,E)$ as $ \irr(G) = \sum_{uv\in E}|d_G(u)-d_G(v)|,$ where $d_G(u)$ denotes the degree of a vertex $u \in V$. Recently, this graph invariant gained interest in the…
Graph operations are crucial for building complicated network architectures from simple graphs. In [7], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and…
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…
The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the…
For a graph $G$, the edge Mostar index of $G$ is the sum of $|m_u(e|G)-m_v(e|G)|$ over all edges $e=uv$ of $G$, where $m_u(e|G)$ denotes the number of edges of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for…
In this paper, we establishe the extremal bounds of the topological indices -- Sigma index -- focusing on analyzing the sharp upper bounds and the lower bounds of the Sigma index, which is known $\sigma(G)=\sum_{uv\in…
In this paper we introduce a variation of the well-known Zagreb indices by considering a proper vertex colouring of a graph $G$. The chromatic Zagreb indices are defined in terms of the parameter $c(v), v \in V(G)$ instead of the invariant…
In this paper, the investigates Adriatic indices, specifically the sum lordeg index where it defined as $SL(G) = \sum_{u \in V(G)} \deg_G(u) \sqrt{\ln \deg_G(u)}$ and the variable sum exdeg index $SEI_a(G)$ for $a>0$, $a\neq 1$. We present…
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…
In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$,…
Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $\Psi(d_{u},d_{v})$, where $uv$ is the edge connecting the vertices $u,v$ of the graph…