First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Abstract

a b s t r a c tLet 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 colored digraphs with exactly one nonsingular cycle

“…Here G b is not necessarily unicyclic. Hence, the following result is a generalization of [8,Theorem 2.14]. Proof.…”

“…In [8], the authors proved that multiplicity of λ 1 (L(G b ) is at most two when G b is unicyclic. Here G b is not necessarily unicyclic.…”

“…eigenvalue over all such graphs. The unique mixed graph which minimizes the smallest Laplacian eigenvalue over all nonsingular unicyclic mixed graphs on n vertices was determined in [8].…”

The Signless Laplacian or Adjacency Spectral Radius of Bicyclic Graphs with Given Number of Cut Edges

“…The least signless Laplacian eigenvalues is also studied; see e.g. [6,17]. Other work can be found in [16,34] for the spectral integrality, [29,35] for the isospectral problem, and [28] for the spectral spread.…”

The least eigenvalue of the signless Laplacian of non-bipartite unicyclic graphs with k pendant vertices

“…Cardoso et al [2] determined the unique graph with the minimum value of the least eigenvalue of the signless Laplacian of a connected non-bipartite graph with a prescribed number of vertices. Fan et al [6] minimized the least eigenvalue among all nonsingular unicyclic mixed graphs in the setting of Laplacian of mixed graphs. Their result can be applied to signless Laplacian of graphs directly.…”