The Spectral Properties of $p$-Sombor (Laplacian) Matrix of Graphs
Received:May 20, 2022  Revised:November 24, 2022
Key Words: $p$-Sombor matrix   $p$-Sombor Laplacian matrix   $p$-Sombor spectrum  
Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11971180; 12271337) and the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No.2022KTSCX225).
Author NameAffiliation
Hechao LIU School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China 
Lihua YOU School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China 
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Abstract:
      The Sombor index, which was recently introduced into chemical graph theory, can predict physico-chemical properties of molecules. In this paper, we investigate the properties of ($p$-)Sombor index from an algebraic viewpoint. The $p$-Sombor matrix $\mathcal{S}_{p}(G)$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $((d_{i})^{p}+(d_{j})^{p})^{\frac{1}{p}}$ if $v_{i}\sim v_{j}$, and 0 otherwise, where $d_{i}$ denotes the degree of vertex $v_{i}$ in $G$. The matrix generalizes the famous Zagreb matrix $(p=1)$, Sombor matrix $(p=2)$ and inverse sum index matrix $(p=-1)$. In this paper, we find a pair of $p$-Sombor noncospectral equienergetic graphs and determine some bounds for the $p$-Sombor (Laplacian) spectral radius. Then we describe the properties of connected graphs with $k$ distinct p-Sombor Laplacian eigenvalues. At last, we determine the Sombor spectrum of some special graphs. As a by-product, we determine the spectral properties of Sombor matrix $(p=2)$, Zagreb matrix $(p=1)$ and inverse sum index matrix $(p=-1)$.
Citation:
DOI:10.3770/j.issn:2095-2651.2023.03.003
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