2025/08/13 by Li, Shuchao, Zhang, Jiaqi
#05C50 #05C75 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.09770
For a given graph G, let A(G), Q(G), and D(G) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of G, respectively. The Aσ(G) matrix, proposed by Nikiforov, is defined as Aσ(G)=σD(G)+(1 - σ)A(G), where σ∈[0,1]. This matrix captures the gradual transition from A(G) to Q(G). Let Gn,α denote the family of all connected graphs with n vertices and independence number α. A graph in Gn,α is referred to as an Aσ-minimizer graph if it achieves the minimum Aσ spectral radius. In this paper, we first demonstrate that the Aσ-minimizer graph in Gn,α must be a tree when α≥\lceil(n)/(2)\rceil, and we provide several characterizations of such Aσ-minimizer graphs. We then specifically characterize the Aσ-minimizer graphs for the case α= \lceil(n)/(2)\rceil + 1. Furthermore, we obtain a structural characterization for the Aσ-minimizer graph when α=n - c, where c≥4 is an integer. Finally, we identify 17 potential Aσ-minimizer graphs within Gn,n - 4, thereby extending the results of Liu and Wang [9].