Related papers: On the first Banhatti-Sombor index

The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d(u)^{2}+d(v)^{2}$ over all edges $uv$ of $G$ , where $d(u)$ denotes the degree of a vertex $u$. In this paper, we give sharp upper bounds…

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…

In a study on the structure--dependency of the total $\pi$-electron energy from 1972, Trinajsti\'c and one of the present authors have shown that it depends on the sums $\sum_{v\in V}d(v)^2$ and $\sum_{v\in V}d(v)^3$, where $d(v)$ is the…

Recently, Gutman [MATCH Commun. Math. Comput. Chem. 86 (2021) 11-16] defined a new graph invariant which is named the Sombor index $\mathrm{SO}(G)$ of a graph $G$ and is computed via the expression \[ \mathrm{SO}(G) = \sum_{u \sim v}…

Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points…

Continuing the recent work of L. Zhong and K. Xu [MATCH Commun. Math. Comput. Chem.71(2014) 627-642], we determine inequalities among several vertex-degree-based topological indices; first geometric-arithmetic index(GA), augmented Zagreb…

In this paper, we presented a study of topological indices on trees, where we show a relationship with irregularity of Albertson index and minimum, maximum degrees $\delta,\Delta$ of graph $G$, where contribute vital roles in determining…

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected…

The atom-bond connectivity (ABC) index is a degree-based topological index. It was introduced due to its applications in modeling the properties of certain molecular structures and has been since extensively studied. In this note, we…

Recently in 2021, Gutman introduced the Sombor index of a graph, a novel degree-based topological index. It has been shown that the Sombor index efficiently models the thermodynamic properties of chemical compounds. Assume $\mathbb{B}_n^k$…

For a graph $G$ without isolated vertices, the inverse degree of a graph $G$ is defined as $ID(G)=\sum_{u\in V(G)}d(u)^{-1}$ where $d(u)$ is the number of vertices adjacent to the vertex $u$ in $G$. By replacing $-1$ by any non-zero real…

The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is…

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index}…

In the paper [I. Gutman, N. Trinajsti\'c, Chem. Phys. Lett. 17 (1972), 535], it was shown that total $\pi$-electron energy ($E$) of a molecule $M$ depends on the quantity $\sum_{v\in V(G)}d_{v}^{2}$ (nowadays known as the "first Zagreb…

Let $G=(V_G, E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^{d}(G) = \sum_{v\in V_G}\varepsilon_{G}(v)D_{G}(v)$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v) =…

In this paper, we have studied bounds based on topological indicators, from which we selected Albertson index $\mathrm{irr}$ and the Sigma index $\sigma$. The Sigma index was defined through the following relationship: \[…

The {\em atom-bond connectivity (ABC) index} is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph $G$, the ABC index is defined as $\sum_{uv\in…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{v_{i}}$ (or $d_{i}$ for short) the degree of vertex $v_{i}$. The $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$ ($p\neq0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is…

Topological indices are numerical invariants derived from molecular graphs and play an important role in characterizing chemical compounds and predicting their properties. Among the earliest descriptors are the classical Zagreb indices…