文献类型：期刊文献

英文题名：On the rank (nullity) of a connected graph?

作者：Wang, Zhi-Wen[1]; Guo, Ji-Ming[1]

机构：[1] Department of Mathematics, East China University of Science and Technology, Shanghai, China

年份：2019

外文期刊名：arXiv

收录：EI(收录号：20200468252)

语种：英文

外文关键词：Eigenvalues and eigenfunctions

摘要：The rank r(G) of a graph G is the rank of its adjacency matrix A(G) and the nullity η(G) of G is the multiplicity of 0 as an eigenvalue of A(G). In this paper, we prove that if G is a connected graph of order n with rank r, then G contains a nonsingular connected induced subgraph of order r. As an application of the result, we completely solve the following problem posed by Zhou, Wong and Sun in [Linear Algebra and its Applications, 555 (2018) 314-320]: Let G be a connected graph of order n with nullity η(G) and the maximum degree ?. Then η(G) ≤ (? ??2)?n1+ 2 , the equality holds if and only if G ～= Cn (n ≡ 0 (mod 4)) or G ～= K?,?. Copyright ? 2019, The Authors. All rights reserved.