Article

The spread of unicyclic graphs with given size of maximum matchings

Nankai University, T’ien-ching-shih, Tianjin Shi, China
Journal of Mathematical Chemistry (Impact Factor: 1.27). 11/2007; 42(4):775-788. DOI: 10.1007/s10910-006-9141-6

ABSTRACT The spread s(G) of a graph G is defined as s(G)=max The spread s(G) of a graph G is defined as s(G)=max
i,j i,j
|λ |λ
i i
−λ −λ
j j
|, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U
*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U *(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U
*(n,k), and the graph with the largest spread in U(n,k). *(n,k), and the graph with the largest spread in U(n,k).

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    • "However, the graph(s) with maximum spread is still unknown, and some conjectures are presented in their paper. In [10], Li, Zhang and Zhou determine the unique graph with maximum spread among all unicyclic graphs with given order not less than 18, which is obtained from a star by adding an edge between two pendant vertices. In [11] Bolian Liu and Muhuo Liu obtain some new lower and upper bounds for the spread of a graph, which are some improvements of Gregory's bound on the spread for graphs with additional restrictions. "
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    ABSTRACT: The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.
    The electronic journal of combinatorics 01/2009; 16. · 0.57 Impact Factor
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    • "Many articles have recently appeared about the eigenvalues of unicyclic graphs in the context of matchings. Topics for such articles include general analysis [1] [2], the spectral radius [3], energy [4] [5], largest eigenvalues [6], and nullity [7]. One paper [8] uses methods similar to those presented here to determine the nullity of a unicyclic graph, but not the number of positive and negative eigenvalues. "
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    ABSTRACT: The inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs.
    Linear Algebra and its Applications 08/2008; 429(4):849-858. DOI:10.1016/j.laa.2008.04.013 · 0.98 Impact Factor
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    • "For trees, it is obvious that the star and the path have the maximum spread and the minimum spread among all trees with given order, respectively. For unicyclic graphs, Li et al. [7] determine the unique graph with maximum spread among all unicyclic graphs with given size of maximum matchings, and the unique graph with maximum spread among all unicyclic graphs of order n when n 18. "
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    ABSTRACT: The spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. Let denote the set of connected unicyclic graphs of order n and girth k, and let Un denote the set of connected unicyclic graphs of order n. In this paper, we determine the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . We, finally, characterize the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in Un.
    Linear Algebra and its Applications 07/2008; 429(2-3-429):577-588. DOI:10.1016/j.laa.2008.03.012 · 0.98 Impact Factor