A new kind of $A_{\p}$-matrix for mixed graphs

Abstract

Nikiforov\cite{nikiforov2017merging} introduced the concept of the $A_{\p}$-matrix as a convex linear combination of a graph's adjacency matrix and its diagonal matrix of vertex degrees. In this paper, we introduce a new variant of the $A_{\p}$-matrix, which is a linear combination of the Hermitian adjacency matrix of the second kind, $H^{\om}$, and the degree-diagonal matrix. This study provides new insights into the $A_{\p}$-matrix as a Hermitian matrix, leading to novel bounds in spectral graph theory.