On the relationship between shortlex order and Aα-spectral radii of graphs with starlike branch tree

Abstract

For graph G, let ρα(G) denote its Aα-spectral radius and let Gv(n1,n2,…,nd) be the graph obtained from G by appending d paths on n1,n2,…,nd vertices respectively at the vertex v of G. For connected graph G with at least two vertices, the famous Li-Feng Grafting Theorem indicates that ρ0(Gv(p+1,q−1))<ρ0(Gv(p,q)) holds for any p≥q≥1. Later, Cvetković and Simić showed that the similar inequality also holds for the signless Laplacian spectral radius (equivalently, the A0.5-spectral radius). Confirming a Nikiforov-Rojo Conjecture, Lin, Huang and Xue (independently, Guo and Zhou) discovered that this phenomenon happens for Aα-spectral radius if 0≤α<1. Recently, Oliveira, Stevanović and Trevisan proved that the spectral radius has similar property for graphs obtained by appending at least three paths or a varying number of paths to an isolated vertex. In this note, we extend these former results by showing that the Aα-spectral radius of Gv(n1,n2,…,nd) will increase according to the shortlex ordering of (n1,n2,…,nd).