Yi-Zheng Fan

Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order

For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the spectral radius of the unoriented Laplacian matrix over all bicyclic graphs of order n, namely, the graph obtained from the cycle C 4 by first adding a chord and then attaching n − 4 pendant edges to one end of the chord.

On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs ☆

Abstract In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph G, we construct a generalized power of G, denoted by G k , s , which is obtained from G by blowing up each vertex into a s-set and each edge into a ( k − 2 s ) -set, where s ≤ k / 2 . When s k / 2 , G k , s is always odd-bipartite. We show that G k , k 2 is non-odd-bipartite if and only if G is non-bipartite, and find that G k , k 2 has the same adjacency (respectively, signless Laplacian) spectral radius as G. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph G hold for G k , k 2 . In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs G k , k 2 of fixed order, and prove that 2 + 5 is the smallest limit point of the non-odd-bipartite hypergraphs G k , k 2 . In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.

On spectral integral variations of mixed graphs

In this paper, we characterize the mixed graphs with exactly one Laplacian eigenvalue moving up by an integer and other Laplacian eigenvalues remaining invariant when an edge is added. The results extend those of Fan [Linear and Multilinear Algebra 50 (2002) 133] for general graphs, and So [Linear and Multilinear Algebra 46 (1999) 193] for simple graphs.

Maximum Estrada index of bicyclic graphs

Let G be a simple graph of order n , let λ 1 ( G ) , λ 2 ( G ) , ? , λ n ( G ) be the eigenvalues of the adjacency matrix of G . The Estrada index of G is defined as E E ( G ) = ? i = 1 n e λ i ( G ) . In this paper we determine the unique graph with maximum Estrada index among bicyclic graphs with fixed order.

On the least eigenvalue of a unicyclic mixed graph

Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.

A note on the nullity of unicyclic signed graphs

Abstract In this paper we introduce the nullity of signed graphs, and give some results on the nullity of signed graphs with pendant trees. We characterize the unicyclic signed graphs of order n with nullity n - 2 , n - 3 , n - 4 , n - 5 respectively.

Maximizing Spectral Radii of Uniform Hypergraphs with Few Edges

Abstract In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic hypergraphs, linear or power unicyclic hypergraphs with given girth, linear or power bicyclic hypergraphs, respectively.

The connectivity and the Harary index of a graph

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with the maximum Harary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among all graphs with given vertex connectivity.

Quadratic forms on graphs with application to minimizing the least eigenvalue of signless Laplacian over bicyclic graphs

Given a graph and a vector defined on the graph, a quadratic form is defined on the graph depending on its edges. In order to minimize the quadratic form on trees or unicyclic graphs associated with signless Laplacian, the notion of basic edge set of a graph is introduced, and the behavior of the least eigenvalue and the corresponding eigenvectors is investigated. Using these results a characterization of the unique bicyclic graph whose least eigenvalue attains the minimum among all non-bipartite bicyclic graphs of fixed order is obtained.