A note on the Q-integrability of complete bipartite graphs with self-loops
Year: 2024
Type: article
Abstract: A graph G is Q-integral if its signless Laplacian has only integral eigenvalues. Suppose G^H is the graph after adding a self-loops to every vertex of a subset H of V(G). In this article, we prove that for a non-empty proper subset H of V(K_{p,p}), $K_{p,p}^{H}$ is Q-integral if and only if either p = 4 and self-loops are added in exactly two vertices of each partite set, or p=2 and self-loops are added to all the vertices of one partite set. Besides, we prove that if H is a non-empty proper subset of V(K_{1,p}), then K_{1,p}^{H} is Q-integral if and only if p=4 and self-loops are added to all the pendant vertices. Using these results, we also answer when the matrix M + 3D' has only integral spectrum, where D' is any $(0,1)$-diagonal matrix and the matrix M is permutation similar to either \begin{pmatrix} O_{p} & J\\ J & O_{p} \end{pmatrix}$ or $\begin{pmatrix} p & J \\ J & I_{p} \end{pmatrix}$.
Authors: Jesmina Pervin, Lavanya Selvaganesh
Institution: Indian Institute of Technology BHU
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4404328694