The Majorization Theorems of Single-Cone Trees and Single-Cone Unicyclic Graphs

A single-cone tree (unicyclic graph) is the join of a complete graph \(K_1\) and a tree (unicyclic graph). Suppose \(\pi =(d_1, d_2, \ldots , d_n)\) and \(\pi ^{\,\prime }=(d_1^{\,\prime }, d_2^{\,\prime }, \ldots , d_n^{\,\prime })\) are two non-increasing degree sequences. We say \(\pi \) is majorizated by \(\pi ^{\,\prime }\), denoted by \(\pi \lhd \pi ^{\,\prime }\), if and only if \(\pi \ne \pi ^{\,\prime }\), \(\sum \nolimits _{i=1}^{n} d_i=\sum \nolimits _{i=1}^{n} d_i^{\,^{\,\prime }}\), and \(\sum \nolimits _{i=1}^j d_i\le \sum \nolimits _{i=1}^j d_i^{\,^{\,\prime }}\) for all \(j=1, 2, \ldots , n-1\). We use \(J_{\pi }\) to denote the class of single-cone trees (unicyclic graphs) with degree sequence \(\pi \). Suppose that \(\pi \) and \(\pi ^{\,\prime }\) are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let \(\rho \) and \(\rho ^{\,\prime }\) be the largest spectral radius of the graphs in \(J_{\pi }\) and \(J_{\pi ^{\,\prime }}\), respectively, \(\mu \) and \(\mu ^{\,\prime }\) be the largest signless Laplacian spectral radius of the graphs in \(J_{\pi }\) and \(J_{\pi ^{\,\prime }}\), respectively. In this paper, we prove that if \(\pi \lhd \pi ^{\,\prime }\), then \(\rho <\rho ^{\,\prime }\) and \(\mu <\mu ^{\,\prime }\).