On the least signless Laplacian eigenvalue of a $P_t$-free non-bipartite connected graph
LIU Xiao-rong1,2, GUO Shu-guang2, ZHANG Rong2
1. Department of Mathematics, Qinghai Normal University, Xining 810008, China
2. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China
Abstract For a connected graph $G$, the least eigenvalue $q_n(G)$ of the signless Laplacian of $G$ equals zero if and only if $G$ is bipartite. $q_n(G)$ is often used to measure the non-bipartiteness of a graph $G$, and has attracted the interest of more and more researchers. This paper investigates conditions depending on $q_n(G)$ under which a graph $G$ contains a long path, and characterizes the extremal graph in which the least signless Laplacian eigenvalue attains the minimum among all the $P_t$-free non-bipartite unicyclic graphs and $P_t$-free non-bipartite connected graphs of order $n$, respectively.
LIU Xiao-rong, GUO Shu-guang, ZHANG Rong. On the least signless Laplacian eigenvalue of a $P_t$-free non-bipartite connected graph. Applied Mathematics A Journal of Chinese Universities, 2015, 30(4): 462-468.
