Spectral Sufficient Conditions on Pancyclic Graphs

A pancyclic graph of order n is a graph with cycles of all possible lengths from 3 to n . In fact, it is NP-complete that deciding whether a graph is pancyclic. Because the spectrum of graphs is convenient to be calculated, in this study, we try to use the spectral theory of graphs to study this problem and give some sufficient conditions for a graph to be pancyclic in terms of the spectral radius and the signless Laplacian spectral radius of the graph.

On the skew Laplacian spectral radius of a digraph

Let [Formula: see text] be an orientation of a simple graph [Formula: see text] with [Formula: see text] vertices and [Formula: see text] edges. The skew Laplacian matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix with oriented degrees [Formula: see text] as diagonal entries and [Formula: see text] is the skew matrix of the digraph [Formula: see text]. The largest eigenvalue of the matrix [Formula: see text] is called skew Laplacian spectral radius of the digraph [Formula: see text]. In this paper, we study the skew Laplacian spectral radius of the digraph [Formula: see text]. We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph [Formula: see text], in terms of different structural parameters of the digraph and the underlying graph. We characterize the extremal digraphs attaining these bounds in some cases. Further, we end the paper with some problems for the future research in this direction.