Aα-Spectrum of a Firefly Graph

Abstract

Let G be a connected graph of order n, A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the row-sums of A(G). In 2017, Nikiforov [Nikiforov, V., Merging the A- and Q-Spectral Theories, Applicable Analysis and Discrete Mathematics 11 (2017), pp. 81–107.] defined the convex linear combinations A_α(G) of A(G) and D(G) by$A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right),0\le \alpha \le 1.$ In this paper, we obtain a partial factorization of the A_α-characteristic polynomial of the firefly graph which explicitly gives some eigenvalues of the graph.