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摘要
Given a graph G, for a real number alpha is an element of[0, 1], Nikiforov (2017) proposed the A alpha-matrix of Gas A alpha(G) = alpha D(G) + (1 - alpha)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of A alpha(G) is called the A alpha-index of G. For n > k, let S-n,S-k be the join of a clique on kvertices with an independent set of n - k vertices. Then S-n,k(+) denotes the graph obtained from S-n,S-k by adding an edge to connect two vertices of degree k. Very recently, Cioab.a, Desai and Tait resolved the Nikiforov's conjecture: For fixed k >= 2, and sufficiently large n, the {C-2k+1, C-2k+2}-free graph of order nwith maximum adjacency spectral radius is S-n,S-k and the C2k+2-free graph of order nwith maximum adjacency spectral radius is S-n,k(+). In this paper, we confirm the A alpha-spectral version of this conjecture: For fixed k >= 2, 0 < alpha < 1and n >= 324k(6)(k+1)2|alpha(6)(1-alpha)(2), the {C2k+1, C2k+2}-free graph of order nwith maximum Aa-index is Sn,kand the C2k+2-free graph of order nwith maximum Aa-index is S+ n,k.
