Some results on the majorization theorem of connected graphs

Abstract

Let π = (d1, d2, ..., dn) and π′ = (d′1, d′2, ..., d′n) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σi=1ndi = Σi=1nd′i, and Σi=1jdi ≤ Σi=1jd′i for all j = 1, 2, ..., n. Weuse Cπ to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Biyikoglu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in Cπ and C′π, respectively, then ρ(G) < ρ(G′).