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For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$SO_{\alpha}(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^\alpha$$ where $SO(G)=SO_{0.5}(G)$ is the…

The sigma index of a graph, defined as the population variance of its degree sequence, is a fundamental measure of structural irregularity. In this paper, we introduce and systematically investigate its natural extension to fuzzy graphs,…

Recently, Gutman defined a new vertex-degree-based graph invariant, named the Sombor index $SO$ of a graph $G$, and is defined by $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2},$$ where $d_G(v)$ is the degree of the vertex $v$ of $G$. In…

The general Sombor index of $G$ is defined as $SO_{\alpha}(G)= \sum_{uv\in G}\left(d^2_{G}(u)+d^2_{G}(v)\right)^{\alpha}$. For $0<\alpha<1$, we have the upper bound of $SO_{\alpha}(G)$ on unicyclic graphs with a fixed diameter, and the…

Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we…

Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as $$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$ in which $d(x)$ is the degree of the…

Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph $G$ as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}=d_{G}(u)$ denotes the degree of vertex $u$ in graph $G$.…

The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree,…

The Sombor index (SO) is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices of $\sqrt{d_i^2+d_j^2}$, where $d_i$ is the degree of the $i$-th vertex. It has been conceived using geometric…

The Sombor index of a graph $G$ was recently introduced by Gutman from the geometric point of view, defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d(u)^2+d(v)^2}$, where $d(u)$ is the degree of a vertex $u$. For two real numbers $\alpha$ and…

In this paper, we investigate the Diminished Sombor index (DSO), a recently introduced degree-based topological index for a simple graph $G$, defined as \[ DSO(G) = \sum_{uv \in E} \frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}, \] where $d_u$ denotes…

The Sombor index is a topological index in graph theory defined by Gutman in 2021. In this article we find the maximum Sombor index of unicyclic graphs with a fixed number of pendant vertices. We also provide the unique graph among the…

We introduce a degree-based variable topological index inspired on the power (or generalized) mean. We name this new index as the mean Sombor index: $mSO_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha+d_v^\alpha \right) /2…

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 $\mathbb{B}_n^k$…

Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let…

The Sombor index is one of the geometry-based descriptors, which was defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d^{2}(u)+d^{2}(v)},$$ where $d(u)$ (resp. $d(v)$) denotes the degree of vertex $u$ (resp. $v$) in $G$. In this note, we determine…

Recently, a novel topological index, Sombor index, was introduced by Gutman, defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. In this paper, we first determine the…

In 2021, the Sombor index was introduced by Gutman, which is a new degree-based topological molecular descriptors. The Sombor index of a graph $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the…

The Sombor index of a graph $G$, introduced by Ivan Gutman, is defined as the sum of the weights $\sqrt{d_G(u)^2+d_G(v)^2}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of vertex $u$ in $G$. The Sombor coindex is recently…

A new geometric background of graph invariants was introduced by Gutman, of which the simplest is the second Sombor index $SO_2$, defined as $SO_2=SO_2(G)=\sum_{uv\in E}\frac{|d^2_G(u)-d^2_G(v)|}{d^2_G(u)+d^2_G(v)}$, where $G = (V, E)$ is a…