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

Abstract

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 $$\mu <\mu ^{\,\prime }$$.