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 π to denote the class of connected graphs with degree sequence π. 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 π . In this paper, we prove that: Let π and π′ 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 π and C π', 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).
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Liu, B., Liu, M. & You, Z. The Majorization Theorem for Signless Laplacian Spectral Radii of Connected Graphs. Graphs and Combinatorics 29, 281–287 (2013). https://doi.org/10.1007/s00373-011-1106-2
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DOI: https://doi.org/10.1007/s00373-011-1106-2