The spectral radius of graphs with given independence number

Graphs with the minimum spectral radius for given independence number

“…1) 2009 Xu, Hong, Shu and Zhai [9] 3 P( n 3 , 3) for 3|n, where W n if n is odd D n if n is even (see Fig. 1) 2009 Xu, Hong, Shu and Zhai [15] n − 4 Theorem 1.2 2022 Lou and Guo [20] n − 3 Theorem 3.2 2009 Xu, Hong, Shu and Zhai [20] n − 2 T (⌈ n−3 2 ⌉, ⌊ n−3 2 ⌋) (see Fig. 1) 2009 Xu, Hong, Shu and Zhai [20] n − 1 the star K 1,n−1 2009 Xu, Hong, Shu and Zhai four lines list the results for small independence number α ∈ {1, 2, 3, 4}, and the others for large α ∈ {⌈ n 2 ⌉, ⌈ n 2 ⌉ + 1, n − 4, n − 3, n − 2, n − 1}.…”

“…In this paper, we restrict on α ≥ ⌈ n 2 ⌉ to characterize the minimizer graph T * and determine its spectral radius ρ(T * ). Recently, for α ≥ ⌈ n 2 ⌉, Lou and Guo [15] gave a general result that the graph with minimum spectral radius in G n,α is a tree. Based on this result, we further give the structural features for the minimizer graph, and then provide of a constructing theorem for it.…”

“…In Section 4, we give a construction theorem for the minimizer graphs in G n,α together with their spectral radii for k = n − α ≤ n 2 ( see Theorem 4.1). In Section 5, as an application, we determine the minimizer graphs in G n,α for α = n − 1, n − 2, n − 3, n − 4, n − 5, n − 6 along with their spectral radii, the first four results are known in [15,20] and the last two are new.…”

“…In this section, we first cite several lemmas in [15] that describe the structure and properties for minimizer graph, and then we provide a series of propositions which together give some new characters of the minimizer graph's structure. We first quote a nice result from [15] which determines the shape of the minimizer graph.…”

“…pointed that determining minimizer graph in G n,α appears to be a tough problem on page 96. Very recently, Lou and Guo in [15] proved that the minimizer graph of G n,α must be a tree if α ≥ ⌈ n 2 ⌉. In this paper, we further give the structural features for the minimizer graph in detail, and then provide of a constructing theorem for it.…”