The [formula omitted]-spectral radius of nonregular graphs (digraphs) and maximum degree (outdegree)

Abstract

For 0≤α<1, let Aα(G)=αD(G)+(1−α)A(G) be the Aα-matrix of G, where D(G) is the diagonal degree matrix of G and A(G) is the adjacency matrix of G Let λα(G) denote the Aα-spectral radius of a graph G. Let G be a k-connected nonregular graph with n vertices, m edges, maximum degree Δ and minimum degree δ. In this paper, we show that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−1)2−(Δ−k+1)(n−k−1)+(1−α)nk2,which extends the result on the spectral radius of Xue and Liu (0000) and improves the result on the signless Laplacian spectral radius of Shiu et al. (2017). Furthermore, we also prove that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−Δ+2k−2)(n−δ−1)+k2+(1−α)nk2,which extends the result on the signless Laplacian spectral radius of Ning et al. (2018).Let Γ be a k-strong nonregular digraph of order n, size m, and maximum outdegree Δ+. For 0≤α<1, let Aα(Γ)=αD(Γ)+(1−α)A(Γ) be the Aα-matrix of Γ, where D(Γ) is the diagonal matrix of its vertex outdegrees and A(Γ) is the adjacency matrix of Γ. Denote by λα(Γ) the Aα-spectral radius of a digraph Γ. We prove that Δ+−λα(Γ)>(1−α)(nΔ+−m)k22(nΔ+−m)(n−1)2−(Δ+−k+1)(n−k−1)+(1−α)nk2,which improves the result of Xi and Wang (2020). In the end of the paper, related problem is mentioned.