The Majorization Theorem for Signless Laplacian Spectral Radii of Connected Graphs

Abstract

Let $${\pi=(d_{1},d_{2},\ldots,d_{n})}$$ and $${\pi'=(d'_{1},d'_{2},\ldots,d'_{n})}$$ be two non-increasing degree sequences. We say $${\pi}$$ is majorizated by $${\pi'}$$ , denoted by $${\pi \vartriangleleft \pi'}$$ , if and only if $${\pi\neq \pi'}$$ , $${\sum_{i=1}^{n}d_{i}=\sum_{i=1}^{n}d'_{i}}$$ , and $${\sum_{i=1}^{j}d_{i}\leq\sum_{i=1}^{j}d'_{i}}$$ for all $${j=1,2,\ldots,n}$$ . If there exists one connected graph G with $${\pi}$$ as its degree sequence and $${c=(\sum_{i=1}^{n}d_{i})/2-n+1}$$ , then G is called a c-cyclic graph and $${\pi}$$ is called a c-cyclic degree sequence. Suppose $${\pi}$$ is a non-increasing c-cyclic degree sequence and $${\pi'}$$ is a non-increasing graphic degree sequence, if $${\pi \vartriangleleft \pi'}$$ and there exists some t $${(2\leq t\leq n)}$$ such that $${d'_{t}\geq c+1}$$ and $${d_{i}=d'_{i}}$$ for all $${t+1\leq i\leq n}$$ , then the majorization $${\pi \vartriangleleft \pi'}$$ is called a normal majorization. Let μ(G) be the signless Laplacian spectral radius, i.e., the largest eigenvalue of the signless Laplacian matrix of G. We use C ? to denote the class of connected graphs with degree sequence ?. If $${G \in C_{\pi}}$$ and $${\mu(G)\geq \mu(G')}$$ for any other $${G'\in C_{\pi}}$$ , then we say G has greatest signless Laplacian radius in C ? . In this paper, we prove that: Let ? and ?? be two different non-increasing c-cyclic (c ? 0) degree sequences, G and G? be the connected c-cyclic graphs with greatest signless Laplacian spectral radii in C ? and C ?', respectively. If $${\pi \vartriangleleft \pi'}$$ and it is a normal majorization, then $${\mu(G) < \mu(G')}$$ . This result extends the main result of Zhang (Discrete Math 308:3143---3150, 2008).