$ A_{\alpha} $ matrix of commuting graphs of non-abelian groups

Abstract

For a finite group $ \mathcal{G} $ and a subset $ X\neq \emptyset $ of $ \mathcal{G} $, the commuting graph, indicated by $ G = \mathcal{C}(\mathcal{G}, X) $, is the simple connected graph with vertex set $ X $ and two distinct vertices $ x $ and $ y $ are edge connected in $ G $ if and only if they commute in $ X $. The $ A_{\alpha} $ matrix of $ G $ is specified as $ A_{\alpha}(G) = \alpha D(G)+(1-\alpha) A(G), \; \alpha\in[0, 1] $, where $ D(G) $ is the diagonal matrix of vertex degrees while $ A(G) $ is the adjacency matrix of $ G. $ In this article, we investigate the $ A_{\alpha} $ matrix for commuting graphs of finite groups and we also find the $ A_{\alpha} $ eigenvalues of the dihedral, the semidihedral and the dicyclic groups. We determine the upper bounds for the largest $ A_{\alpha} $ eigenvalue for these graphs. Consequently, we get the adjacency eigenvalues, the Laplacian eigenvalues, and the signless Laplacian eigenvalues of these graphs for particular values of $ \alpha $. Further, we show that these graphs are Laplacian integral.

Keywords

• a_α matrix
• commuting graph
• adjacency matrix
• laplacian matrix
• signless laplacian matrix
• non-abelian groups