2013/11/08 by Zhang, Jie, Zhang, Xiao-Dong
#05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1311.1987
Let G be a graph of order n and let L(G,λ)=∑k=0n (-1)kck(G)λn-k be the characteristic polynomial of its Laplacian matrix. Motivated by Ilić and Ilić's conjecture [A. Ilić, M. Ilić, Laplacian coefficients of trees with given number of leaves or vertices of degree two, Linear Algebra and its Applications 431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian coefficients in the set Un,l of all n-vertex unicyclic graphs with the number of leaves l, we investigate properties of the minimal elements in the partial set (Un,lg, \preceq) of the Laplacian coefficients, where Un,lg denote the set of n-vertex unicyclic graphs with the number of leaves l and girth g. These results are used to disprove their conjecture. Moreover, the graphs with minimum Laplacian-like energy in Un,lg are also studied.