On Distinct Inertias of Lanzhou Matrices Versus Adjacency Matrices in Some Graph Classes

The Lanzhou matrix is a recently introduced graph matrix that depends on the adjacency relation and on the degrees of each vertex in the graph and its complement, thereby providing a spectral perspective on graph structure that differs from the classical adjacency matrix. The inertia of a matrix M is the triplet InM=n+,n−,n0 that counts the number of positive, negative, and zero eigenvalues of M. In this manuscript, we consider graph classes for which the inertia with respect to the Lanzhou matrix differs from that with respect to the adjacency matrix. For several such classes, we provide closed-form expressions for the inertias and spectra of both matrices. We also identify nonregular graphs whose Lanzhou inertia equals the adjacency inertia of certain other graphs. While a complete characterization of these graph classes remains open, our results give a systematic description of when and how the Lanzhou matrix deviates from the spectral inertia pattern of the classical adjacency matrix.