24 Results Found

The concept of Sombor index $\left(SO\right)$ was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index $SO$: $SO=SO\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}\sqrt{d_G{\left(v_i\right)}^2+d_G{\left(v_j\right)}^2},$ where $d_G\left(v_i\right)$ is the degree...

Let G be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as...

The Hyperbolic Sombor index $HSO\left(G\right)$ of a graph $G=\left(V\right(G),E(G\left)\right)$ is proposed as a new vertex-degree-based topological index. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs....

In this paper, motivated by the recently introduced topological indices—the Sombor index, Sombor coindex, and Lanzhou index, we define a new index—the Lanzhou coindex of a graph. Furthermore, we investigate the Sombor index (coindex) and...

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Then, the Sombor index of G is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$. In this paper, we give some lemmas that c...

Let $\mathcal{Z}{\left(\mathcal{R}\right)}^{\prime }$ denote the set of all elements in the ring $\mathcal{R}$ that are neither zero nor units, where $\mathcal{R}$ is assumed to be a commutative ring with a multiplicative identity satisfying $1\ne 0.$ Two distinct vertices w and $\kappa$ are defined to be adjacent...

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed...

The diminished Sombor index is a degree-based topological index that normalizes the Sombor contribution of each edge (defined as the Euclidean norm of the endpoint degrees) by the sum of those degrees, thereby making the index independent of graph si...

Let G be a graph with set of vertices $V\left(G\right)$$\left(\right|V\left(G\right)|=n)$ and edge set $E\left(G\right)$. Very recently, a new degree-based molecular structure descriptor, called Sombor index is denoted by $SO\left(G\right)$ and is defined as $SO=SO\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}\sqrt{d_G{\left(v_i\right)}^2+d_G{\left(v_j\right)}^2}$, where $d_G\left(v_i\right)$ is th...

The Sombor index is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d^2\left(u\right)+d^2\left(v\right)}$, where $d\left(u\right)$ and $d\left(v\right)$ represent the number of edges in the graph G connected to the vertices u and v, respectively. In this paper, we characterize the largest and second largest Sombor...

The Sombor index has recently become a central tool among degree-based graph invariants; however, it does not explicitly isolate degree imbalance along edges. In this work, we introduce the degree-based Contrastive Sombor Index (CSO), which combines...

The Sombor index of a graph G is $SO\left(G\right)={\sum }_{uv\in E\left(G\right)}{(d_G{\left(u\right)}^2+d_G{\left(v\right)}^2)}^{1/2}$, where $d_G\left(v\right)$ denotes the degree of v. The repetition number of G is the maximum multiplicity of an entry in its degree sequence. In this paper, we determine the maximum Sombor...

In this paper, we present the upper and lower bounds on Sombor index $SO\left(G\right)$ among all connected graphs (respectively, connected bipartite graphs). We give some sharp lower and upper bounds on $SO\left(G\right)$ among connected graphs in terms of some parameters, i...

This work explores the unit graph of the commutative ring ${\mathbb{Z}}_p$ for prime $p$ and explores the significance of the Sombor index and the Sombor polynomial in comprehending its structural details. We systematically investigate the unit graph and the identit...

With geometric significance, the Euler Sombor index of a graph $\Gamma$ is defined as $EP(\Gamma )={\sum }_{\left\{uv\right\}\in E(\Gamma )}\sqrt{d{\left(u\right)}^2+d{\left(v\right)}^2+d\left(u\right)d\left(v\right)}$. It originates from the mathematical distance property and has been proven to have good chemical applicatio...

The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the id...

Consider a simple graph G with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A graph invariant for G is a number related to the structure of G, which is invariant under the symmetry of G. The Sombor index of G is a new graph invariant defined as $SO\left(G\right)={\sum }_{uv\&i}$...

Let G be a simple graph with the vertex set $V=\{v_1,\dots ,v_n\}$ and denote by $d_{v_i}$ the degree of the vertex $v_i$. The modified Sombor index of G is the addition of the numbers ${(d_{v_i}^2+d_{v_j}^2)}^{-1/2}$ over all of the edges $v_i{v}_j$ of G. The modified Sombor mat...

Any number that can be uniquely identified and varied by a graph is known as a graph invariant. This paper will talk about three unique variations of bridge networks, sierpinski networks, honeycomb, and hexagonal networks, with great capability of fo...

In this paper, we introduce some new versions based on the locating vectors named locating indices. In particular, Hyper locating indices, Randić locating index, and Sambor locating index. The exact formulae for these indices of some well-known...

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shann...

A connected graph with p vertices and q edges satisfying $q=p+1$ is referred to as a bicyclic graph. This paper is concerned with an optimal study of the BID (bond incident degree) indices of fixed-size bicyclic graphs with a given matching number. Her...

The reciprocal mean square (RMS) index of a graph G with edge set $E\left(G\right)$ is defined by $\mathcal{RMS}\left(G\right)={\sum }_{uv\in E\left(G\right)}{[{\left(d\left(u\right)\right)}^2+{\left(d\left(v\right)\right)}^2]}^{-1},$ where $d\left(u\right)$ and $d\left(v\right)$ are the degrees of vertices u and v, respectively. We first establish several bounds for the R...

Consider a unicyclic graph G with edge set $E\left(G\right)$. Let $\mathfrak{f}$ be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as ${\mathcal{I}}_{\mathfrak{f}}($...