On Aα spectral extrema of graphs forbidding even cycles

Abstract

Given a graph G, for a real number α∈[0,1], Nikiforov (2017) proposed the Aα-matrix of G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of Aα(G) is called the Aα-index of G. For n>k, let Sn,k be the join of a clique on k vertices with an independent set of n−k vertices. Then Sn,k+ denotes the graph obtained from Sn,k by adding an edge to connect two vertices of degree k. Very recently, Cioabǎ, Desai and Tait resolved the Nikiforov's conjecture: For fixed k≥2, and sufficiently large n, the {C2k+1,C2k+2}-free graph of order n with maximum adjacency spectral radius is Sn,k and the C2k+2-free graph of order n with maximum adjacency spectral radius is Sn,k+. In this paper, we confirm the Aα-spectral version of this conjecture: For fixed k≥2,0<α<1 and n≥324k6(k+1)2α6(1−α)2, the {C2k+1,C2k+2}-free graph of order n with maximum Aα-index is Sn,k and the C2k+2-free graph of order n with maximum Aα-index is Sn,k+.