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 Aα(G) of A(G) and D(G) byAα(G)=αD(G)+(1−α)A(G),0≤α≤1. In this paper, we obtain a partial factorization of the Aα-characteristic polynomial of the firefly graph which explicitly gives some eigenvalues of the graph.