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 0
·
0 Bookmarks
·
48 Views
• Source
Article: Bounds on the largest eigenvalues of trees with a given size of matching
[hide abstract]
ABSTRACT: Very little is known about upper bound for the largest eigenvalue of a tree with a given size of matching. In this paper, we find some upper bounds for the largest eigenvalue of a tree in terms of the number of vertices and the size of matchings, which improve some known results.
Linear Algebra and its Applications. 01/2002;
• Source
Article: The Spread of the Spectrum of a Graph
[hide abstract]
ABSTRACT: Upper and lower bounds are obtained for the spread 1 Gamma n of the eigenvalues 1 2 Delta Delta Delta n of the adjacency matrix of a simple graph. MSC: 05C50, 15A42; 15A36 Keywords: Eigenvalues; Spread; Adjacency matrix 1 Introduction For an n Theta n complex matrix M , the spread, s(M ), of M is defined as the diameter of its spectrum: s(M) := max i;j j i Gamma j j, where the maximum is taken over all pairs of eigenvalues of M . There is a considerable literature on the spread of an arbitrary matrix [9, 12, 13, 15]. The following striking theorem is due to Mirsky [12]. Theorem 1.1 s(M) (2 P i;j jm i;j j 2 Gamma 2 n j P i m i;i j 2 ) 1 2 with equality if and only if M is a normal matrix with n Gamma 2 of its eigenvalues all equal to the average of the remaining two. Suppose M is Hermitian. In that case, the eigenvalues i = i (M) of M are real and may always be assumed to be in nonincreasing order: 1 2 Delta Delta Delta n . Then s(M) = 1 G...
10/1999;
• Source
Article: Lower bounds for the spread of a matrix
[hide abstract]
ABSTRACT: A characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread. Comparisons are then made with several known bounds.
Linear Algebra and its Applications. 01/1985;