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arXiv 2108.08552math.CO
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关于图的Sombor特征多项式和Sombor能量

On the Sombor characteristic polynomial and Sombor energy of a graph

Nima Ghanbari

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15 pages, 2 figures, 2 tables
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摘要：

设$G$是一个顶点集为$V(G) = \{v_1, v_2,\ldots, v_n\}$的简单图.由$A_{SO}(G)$表示的$G$的Sombor矩阵被定义为$n\times n$矩阵，如果$v_i$和$v_j$相邻，则其$(i,j)$条目是$\sqrt{d_i^2+d_j^2}$，而对于另一种情况，其$0$条目是$0$。设Sombor矩阵$A_{SO}(G)$的特征值为$ρ_1\geq ρ_2\geq \ldots\geq ρ_n$，其为Sombor特征多项式$\prod_{i=1}^n (ρ-ρ_i)$的根。$G$的Sombor能量$En_{SO}$是$A_{SO}(G)$的本征值的绝对值之和。本文计算了某些图类的Sombor特征多项式和Sombor能量，定义了Sombor能量唯一性，并提出了一个关于Sombor能量的猜想。

英文摘要：

Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$. The Sombor matrix of $G$, denoted by $A_{SO}(G)$, is defined as the $n\times n$ matrix whose $(i,j)$-entry is $\sqrt{d_i^2+d_j^2}$ if $v_i$ and $v_j$ are adjacent and $0$ for another cases. Let the eigenvalues of the Sombor matrix $A_{SO}(G)$ be $ρ_1\geq ρ_2\geq \ldots\geq ρ_n$ which are the roots of the Sombor characteristic polynomial $\prod_{i=1}^n (ρ-ρ_i)$. The Sombor energy $En_{SO}$ of $G$ is the sum of absolute values of the eigenvalues of $A_{SO}(G)$. In this paper we compute the Sombor characteristic polynomial and the Sombor energy for some graph classes, define Sombor energy unique and propose a conjecture on Sombor Energy.

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