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Spectra of extended double cover graphs

Zhibo Chen — 2004

Czechoslovak Mathematical Journal

The construction of the extended double cover was introduced by N. Alon [1] in 1986. For a simple graph G with vertex set V = { v 1 , v 2 , , v n } , the extended double cover of G , denoted G * , is the bipartite graph with bipartition ( X , Y ) where X = { x 1 , x 2 , , x n } and Y = { y 1 , y 2 , , y n } , in which x i and y j are adjacent iff i = j or v i and v j are adjacent in G . In this paper we obtain formulas for the characteristic polynomial and the spectrum of G * in terms of the corresponding information of G . Three formulas are derived for the number of spanning trees in G * for a connected...

On integral sum graphs with a saturated vertex

Zhibo Chen — 2010

Czechoslovak Mathematical Journal

As introduced by F. Harary in 1994, a graph G is said to be an i n t e g r a l s u m g r a p h if its vertices can be given a labeling f with distinct integers so that for any two distinct vertices u and v of G , u v is an edge of G if and only if f ( u ) + f ( v ) = f ( w ) for some vertex w in G . We prove that every integral sum graph with a saturated vertex, except the complete graph K 3 , has edge-chromatic number equal to its maximum degree. (A vertex of a graph G is said to be if it is adjacent to every other vertex...

Interpolation theorem for a continuous function on orientations of a simple graph

Fu Ji ZhangZhibo Chen — 1998

Czechoslovak Mathematical Journal

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O 1 and O 2 of G , | f ( O 1 ) - f ( O 2 ) | 1 whenever O 1 and O 2 differ in the orientation of exactly one edge of G . We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O 1 and O 2 of G with f ( O 1 ) = p and f ( O 2 ) = q , where p < q , then for any integer k such that p < k < q , there are...

Limit points of eigenvalues of (di)graphs

Fu Ji ZhangZhibo Chen — 2006

Czechoslovak Mathematical Journal

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph D , the set of limit points of eigenvalues of iterated subdivision digraphs of D is the unit circle in the complex plane if and only if D has a directed cycle. 3. Every limit point of eigenvalues of a set...

On graphs with the largest Laplacian index

Bo Lian LiuZhibo ChenMuhuo Liu — 2008

Czechoslovak Mathematical Journal

Let G be a connected simple graph on n vertices. The Laplacian index of G , namely, the greatest Laplacian eigenvalue of G , is well known to be bounded above by n . In this paper, we give structural characterizations for graphs G with the largest Laplacian index n . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k -regular graph G of order n with the largest Laplacian...

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