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Laplacian coefficients of unicyclic graphs with the number of leaves and girth

Abstract

Let GG be a graph of order nn and let L(G,Ξ»)=βˆ‘k=0n(βˆ’1)kck(G)Ξ»nβˆ’k\mathcal{L}(G,\lambda)=\sum_{k=0}^n (-1)^{k}c_{k}(G)\lambda^{n-k} be the characteristic polynomial of its Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c}, M. Ili\'{c}, 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\mathcal{U}_{n,l} of all nn-vertex unicyclic graphs with the number of leaves ll, we investigate properties of the minimal elements in the partial set (Un,lg,βͺ―)(\mathcal{U}_{n,l}^g, \preceq) of the Laplacian coefficients, where Un,lg\mathcal{U}_{n,l}^g denote the set of nn-vertex unicyclic graphs with the number of leaves ll and girth gg. These results are used to disprove their conjecture. Moreover, the graphs with minimum Laplacian-like energy in Un,lg\mathcal{U}_{n,l}^g are also studied.Comment: 19 page, 4figure

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