On the α-index of graphs with pendent paths


Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], write Aα(G) for the matrix Aα(G)=αD(G)+(1−α)A(G). This paper presents some extremal results about the spectral radius ρα(G) of Aα(G) that generalize previous results about ρ0(G) and ρ1/2(G). In particular, write Bp,q,r be the graph obtained from a complete graph Kp by deleting an edge and attaching paths Pq and Pr to its ends. It is shown that if α∈[0,1) and G is a graph of order n and diameter at least k, then ρα(G)≤ρα(Bn−k+2,⌊k/2⌋,⌈k/2⌉), with equality holding if and only if G=Bn−k+2,⌊k/2⌋,⌈k/2⌉. This result generalizes results of Hansen and Stevanović [5], and Liu and Lu [7].

Publication Title

Linear Algebra and Its Applications