On the spread of the generalized adjacency matrix of a graph On the spread of the generalized adjacency matrix of a graph

Abstract

Let  $\mathsf{D}(G)$ and  $\mathsf{A}(G)$, respectively, be the diagonal matrix of vertex degrees and the adjacency matrix of a connected graph $G$. The  generalized adjacency matrix of $G$ is defined as   $\mathsf{A}_{\alpha}(G)=\alpha \mathsf{D}(G)+(1-\alpha)\mathsf{A}(G)$,  $\alpha\in [0,1]$.  The spread of the generalized adjacency matrix, denoted by $\mathsf{S}(\mathsf{A}_{\alpha}(G))$, is defined as the difference of the largest and smallest eigenvalues of $\mathsf{A}_{\alpha}(G)$. The spread $ \mathsf{S}(\mathsf{Q}(G))$  of $\mathsf{Q}(G)$, the signless Laplacian matrix of $G$ and $ \mathsf{S}(\mathsf{A}(G))$ of $\mathsf{A}(G)$ are similarly defined. In this paper, we establish the relationships between $\mathsf{S}(\mathsf{A}_{\alpha}(G))$, $\mathsf{S}(\mathsf{Q}(G))$ and $\mathsf{S}(\mathsf{A}(G))$. Further, we find bounds for $\mathsf{S}(\mathsf{A}_{\alpha}(G))$ involving different parameters of $G$. Also, we obtain lower bounds for $\mathsf{S}(\mathsf{A}_{\alpha}(G))$ in terms of the clique number and independence number of $G$, and we characterize the extremal graphs in some cases.