Abstract
Let \(G = (V(G), E(G))\) be a simple graph with vertex set
\(V(G) = \{v_1 , v_2 , \ldots, v_n\}\) and edge set \(E(G)\).
The \(p\)-Sombor matrix \(S_p(G)\) 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 \(SO_p(G)\) and \(p\)-Sombor matrix \(S_p(G)\)
by the \(k\)-th spectral moment \(N_k\) and the spectral radius
of \(S_p(G)\). 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.