Cospectral graphs obtained by edge deletion
Chris Godsil, Wanting Sun, Xiaohong Zhang
TL;DR
The paper investigates when edge deletions inside cliques of 1-walk-regular graphs yield nonisomorphic graphs that are cospectral for the four standard matrices $A$, $L$, $S$, and $N$ (and their complements). Using a framework based on spectral decompositions, Schur products, and rank-one perturbations (notably Sherman–Morrison), it proves that for cospectral 1-walk-regular graphs $X_1,X_2$ and isomorphic subgraphs $Y_1\cong Y_2$ inside cliques, the edge-deleted graphs $X_1\setminus E(Y_1)$ and $X_2\setminus E(Y_2)$ (and their complements) are cospectral with respect to all four matrices. The results extend to deleting subgraphs in cliques and yield large families of pairwise nonisomorphic cospectral graphs, including substantial collections derived from strongly regular graphs; the unsigned Laplacian case and broader perturbations are also covered. These constructions enrich the catalog of cospectral graphs and connect to applications in quantum information transfer, where spectral invariance under structural perturbations is relevant.
Abstract
Let $M\circ N$ denote the Schur product of two matrices $M$ and $N$. A graph $X$ with adjacency matrix $A$ is walk regular if $A^k\circ I$ is a constant times $I$ for each $k\ge0$, and $X$ is 1-walk-regular if it is walk regular and $A^k\circ A$ is a constant times $A$ for each $k\ge0$. Assume $X$ is 1-walk regular. Here we show that by deleting an edge in $X$, or deleting edges of a graph inside a clique of $X$, we obtain families of graphs that are not necessarily isomorphic, but are cospectral with respect to four types of matrices: the adjacency matrix, Laplacian matrix, unsigned Laplacian matrix, and normalized Laplacian matrix. Furthermore, we show that removing edges of Laplacian cospectral graphs in cliques of a 1-walk regular graph results in Laplacian cospectral graphs; removing edges of unsigned Laplacian cospectral graphs whose complements are also cospectral with respect to the unsigned Laplacian in cliques of a 1-walk regular graph results in unsigned Laplacian cospectral graphs.