An A_α-Spectral Version of the Bhattacharya-Friedland-Peled Conjecture - Yu, Yuantian; Geng, Xianya; Li, Shuchao (The Electronic Journal of Combinatorics, 2024)
In 1985, Brualdi and Hoffman posed the following conjecture: Let G be an n-vertex graph of size m, where 0≤ m≤n2. If m=a2+b with 0≤ b<a, then G(K_b(K_a-bK_1))(n-a-1)K_1 is the unique graph having the largest spectral radius. This conjecture was completely resolved by Rowlinson (1988). In 2018, Bhattacharya, Friedland and Peled posed the bipartite version of Brualdi-Hoffman conjecture. Here, we consider an A_α-spectral extremal question, which may be seen as an A_α-spectral version of the Bhattacharya-Friedland-Peled conjecture: For fixed α[0,1), which graph attains the maximum A_α-index over all bipartite graphs with n vertices and m edges? When 12≤α<1, we prove that for every pair of positive integers n, m, if m=k(n-k), where k is a positive integer with k eq 1,n-1, then the complete bipartite graph K_k,n-k is the unique graph that maximizes the A_α-index over all bipartite graphs with n vertices and m edges; if n≤ m≤ 2n-5, then K_2,n-2^m, the graph obtained from the complete bipartite graph K_2,n-2 by deleting 2n-4-m edges which are incident on a common vertex of degree n-2, is the unique graph that maximizes the A_α-index over all bipartite graphs with n vertices and m edges; if 2n-3≤ m≤ 22(n-4), then K_3,n-3^m, the graph obtained from the complete bipartite graph K_3,n-3 by deleting 3n-9-m edges which are incident on a common vertex of degree n-3, is the unique graph that maximizes the A_α-index over all bipartite graphs with n vertices and m edges. It improves some known ones of Zhang and Li (2017) and partially answer some questions posed by Zhai, Lin and Zhao (2022).