On the signless Laplacian and Laplacian spectral radius of triangle-free $k$-cyclic graphs
HE Chun-yang1,2, GUO Shu-guang2
1. Department of Mathematics, Qinghai Normal University, Xining 810008, China
2. School of Mathematical Sciences, Yancheng Teachers University, Yancheng 224002, China
Abstract A $k$-cyclic graph is a connected graph in which the number of edges equals the number of vertices plus $k+1$. This paper determines the maximal signless Laplacian spectral radius together with the corresponding extremal graph among all triangle-free $k$-cyclic graphs of order $n$. Moreover, this paper gives the first five triangle-free unicyclic graphs on $n \,(n\geq 8)$ vertices, and the first eight triangle-free bicyclic graphs on $n \,(n\geq 12)$ vertices according to the signless Laplacian spectral radius. Finally, the authors of this paper show that the results obtained in this paper also hold for Laplacian spectral radius of triangle-free $k$-cyclic graphs of order $n$.
HE Chun-yang, GUO Shu-guang. On the signless Laplacian and Laplacian spectral radius of triangle-free $k$-cyclic graphs. Applied Mathematics A Journal of Chinese Universities, 2014, 29(3): 295-302.
