Article

# The spread of unicyclic graphs with given size of maximum matchings

Journal of Mathematical Chemistry (Impact Factor: 1.23). 01/2007; 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).

0 Bookmarks
·
68 Views
• ##### Article: P01-518 - Quality of life and associated factors among chronic mental illness patients in Kaohsiung
European Psychiatry - EUR PSYCHIAT. 01/2011; 26:522-522.
• Source
##### Article: The inertia of unicyclic graphs and the implications for closed-shells
[Hide abstract]
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 01/2008; 429(4):849-858. · 0.97 Impact Factor
• ##### Article: The Laplacian spread of quasi-tree graphs
[Hide abstract]
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. 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 quasi-tree graph with maximum Laplacian spread among all quasi-tree graphs in the set Q(n,d) with 1⩽d⩽n-42.
Linear Algebra and its Applications 01/2011; 435(1):60-66. · 0.97 Impact Factor