abs_vol18_pp719-734. 25KB Jun 04 2011 12:05:47 AM

ELA

ON THE CHARACTERIZATION OF GRAPHS WITH PENDENT

VERTICES AND GIVEN NULLITY

∗

BOLIAN LIU†_{, YUFEI HUANG}†_{,AND SIYUAN CHEN}†

Abstract. Let G be a graph with n vertices. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we characterize the graphs (resp. bipartite graphs) with pendent vertices and nullityη, where 0< η≤n. Moreover, the minimum (resp. maximum) number of edges for all (connected) graphs with pendent vertices and nullityηare determined, and the extremal graphs are characterized.

Key words. Eigenvalue, Nullity, Pendent vertex.

AMS subject classifications.05C50.

∗ _{Received by the editors February 28, 2009. Accepted for publication November 11, 2009.}

Handling Editor : Bryan L. Shader.

†_{School of Mathematical Science, South China Normal University, Guangzhou, 510631, P.R.}

China (liubl@scnu.edu.cn, fayger@qq.com, csy me@163.com). The first author is supported by NSF of China (NO.10771080) and SRFDP of China (NO.20070574006).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 719-734, November 2009