The spread of unicyclic graphs with given size of maximum matchings
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 trianglefree 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 trianglefree 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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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 lineartime algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closedshell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closedshell, and hence the reported result gives a lineartime algorithm for determining this for unicyclic graphs.Linear Algebra and its Applications 08/2008; 429(4):849858. DOI:10.1016/j.laa.2008.04.013 · 0.98 Impact Factor 
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ABSTRACT: Authors’ abstract: Let G be a connected graph, and let ϵ b (G) and S Q (G) be the edge bipartiteness and the signless Laplacian spread of G, respectively. We establish some important relationships between ϵ b (G) and S Q (G), and prove S Q (G)≥2(1+cosπ n), with equality if and only if G=P n or G=C n in case of odd n. In addition, we show that if G≠P n or G≠C 2k+1 , then S Q (G)≥4, with equality if and only if G is one of the following graphs: K 1,3 , K 4 , two triangles connected by an edge, and C n for even n. As a consequence, we prove a conjecture of D. Cvetković et al. on minimal signless Laplacian spread [Publ. Inst. Math., Nouv. Sér. 81(95), 11–27 (2007; Zbl 1164.05038)].Applicable Analysis and Discrete Mathematics 04/2012; 6(1). DOI:10.2298/AADM120127003F · 0.71 Impact Factor 
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ABSTRACT: The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the secondsmallest eigenvalue of the Laplacian matrix of the graph. Bao, Tan and Fan [Y.H. Bao, Y.Y. Tan,Y.Z. Fan, The Laplacian spread of unicyclic graphs, Appl. Math. Lett. 22 (2009) 1011–1015.] characterize the unique unicyclic graph with maximum Laplacian spread among all connected unicyclic graphs of fixed order. In this paper, we characterize the unique quasitree graph with maximum Laplacian spread among all quasitree graphs in the set Q(n,d) with 1⩽d⩽n42.Linear Algebra and its Applications 07/2011; 435(1):6066. DOI:10.1016/j.laa.2010.04.013 · 0.98 Impact Factor