Some results involving the Aα-eigenvalues for graphs and line graphs

Abstract

Let GG be a simple graph with adjacency matrix A(G)A\left(G), degree diagonal matrix D(G),D\left(G), and let l(G)l\left(G) be the line graph of GG. In 2017, Nikiforov defined the Aα{A}_{\alpha }-matrix of GG, Aα(G){A}_{\alpha }\left(G), as a linear convex combination of A(G)A\left(G) and D(G)D\left(G), in the following way, Aα(G)≔αA(G)+(1−α)D(G),{A}_{\alpha }\left(G):= \alpha A\left(G)+\left(1-\alpha )D\left(G), where α∈[0,1]\alpha \in \left[0,1]. In this study, we present some bounds for the largest eigenvalue of Aα(G),{A}_{\alpha }\left(G), and for some eigenvalues of Aα(l(G)){A}_{\alpha }\left(l\left(G)). Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.

Keywords

• line graphs
• characteristic polynomial
• aα-eigenvalues
• aα-matrix
• 05c50
• 05c35