24,172 research outputs found

    The Roots and Links in a Class of MM-Matrices

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    In this paper, we discuss exiting roots of sub-kernel transient matrices PP associated with a class of Mβˆ’M- matrices which are related to generalized ultrametric matrices. Then the results are used to describe completely all links of the class of matrices in terms of structure of the supporting tree.Comment: 11 pages, 1 figur

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

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    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

    The Minimum Spectral Radius of Graphs with the Independence Number

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    In this paper, we investigate some properties of the Perron vector of connected graphs. These results are used to characterize that all extremal connected graphs with having the minimum (maximum) spectra radius among all connected graphs of order n=kΞ±n=k\alpha with the independence number Ξ±\alpha, respectively.Comment: 28 pages, 3 figure

    The Wiener and Terminal Wiener indices of trees

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    Heydari \cite{heydari2013} presented very nice formulae for the Wiener and terminal Wiener indices of generalized Bethe trees. It is pity that there are some errors for the formulae. In this paper, we correct these errors and characterize all trees with the minimum terminal Wiener index among all the trees of order nn and with maximum degree Ξ”\Delta.Comment: 13 page
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