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On graphs with integer sombor indices | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 6، دوره 9، شماره 4، اسفند 2024، صفحه 693-705 اصل مقاله (969.27 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28334.1510 | ||
| نویسندگان | ||
| Marzie Sepehr؛ Nader Jafari Rad* | ||
| Department of Mathematics, Shahed University, Tehran, Iran | ||
| چکیده | ||
| 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 $1680$. | ||
| کلیدواژهها | ||
| Integer Sombor index؛ Bipartite graphs؛ $r$-degree&lrm | ||
| مراجع | ||
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