On graphs with integer sombor indices

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Sepehr Marzie; Jafari Rad Nader

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Journal Paper

Paper Information

Journal: COMMUNICATIONS IN COMBINATORICS AND OPTIMIZATION Year:2024 | Volume:9 | Issue:4 Page(s): 693-705

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Information Journal Paper

Title

On graphs with integer sombor indices

Author(s)

Sepehr Marzie | Jafari Rad Nader | Issue Writer Certificate

Keywords

Integer Sombor index

Bipartite graphs

$r$-degree&lrm

Abstract

‎Sombor index of a graph $G$ is defined by $SO(G) = \sum_{uv \in E(G)} \sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ of $G$. An $r$-degree graph is a graph whose degree sequence includes exactly $r$ distinctive numbers. In this article, we study $r$-degree connected graphs with Integer Sombor index for $r \in \{5, 6, 7\}$. We show that: if $G$ is a 5-degree connected graph of order $n$ with Integer Sombor index then $n \geq 50$ and the equality occurs if only if $G$ is a bipartite graph of size 420 with $SO(G) = 14830$; if $G$ is a 6-degree connected graph of order $n$ with Integer Sombor index then $n \geq 75$ and the equality is established only for the bipartite graph of size $1080$; and if $G$ is a 7-degree connected graph of order $n$ with Integer Sombor index then $n \geq 101$ and the equality is established only for the bipartite graph of size $1680$.

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