Spectral properties of $p$-Sombor matrices and beyond

Abstract

Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)=\{v_1,v_2,\cdots ,v_n\}$ and edge set $E(G)$. The $p$-Sombor matrix ${\mathcal{S}}_p(G)$ of $G$ 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 $G$. In this paper, we study the relationship between $p$-Sombor index $S{O}_p(G)$ and $p$-Sombor matrix ${\mathcal{S}}_p(G)$ by the $k$-th spectral moment $N_k$ and the spectral radius of ${\mathcal{S}}_p(G)$. Then we obtain some bounds of $p$-Sombor Laplacian eigenvalues, $p$-Sombor spectral radius, $p$-Sombor spectral spread, $p$-Sombor energy and $p$-Sombor Estrada index. We also investigate the Nordhaus-Gaddum-type results for $p$-Sombor spectral radius and energy. At last, we give the regression model for boiling point and some other invariants.

30 pages, 13 figures

References in corpus (1)

• Sombor Index of Polymers