The Spectral Sufficient Conditions Of Traceability And Pancyclicity Of Graphs

Judging whether a graph is a Hamilton graph is a very important NP-complete problem in graph theory.It has always been widely concerned by graph theory and mathematics workers.Since the spectrum of the graph can reflect the structural properties of the graph well and is easy to calculate.Therefore,study the structural properties of graphs by using spectral graph theory has gradually become a research hotspot in recent years.The adjacent(signless Laplacian)spectral radius of a graph is defined as the maximum eigenvalue of the adjacent(signless Laplacian)matrix corresponding to the graph.The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix,it can be regarded as a graph-spectrum-based invariant.The graph is said to be traceable if it contain a path which contains all vertices.The graph is said to be pancyclic if contains cycles of all possible lengths.This paper mainly studies the traceability and pancyclicity ofgraphs.The detailed contents are as follows:In Chapter 1.firstly,we introduce the background and significance,then in-troduce the concepts,definitions and terminologies involved.Finally we introduce the progress and main conclusions in this paper.In Chapter 2,firstly,we establish some sufficient conditions for a nearly balanced bipartite graph to be traceable in terms of the energy of the quasi-complement of the graph.Subsequently,we introduce the minimum degree con-dition of graphs and characterize the traceability of nearly balanced bipartite graph with minimum degree in terms of the energy.In Chapter 3,firstly,we optimize the edge conditions of pancyclic graph and consider the relationship between edge numbers and extreme spectra.Then,we characterize the pancyclicity of graph and bipancyclicity of balanced bipar-tite graph in terms of the spectral radius and signless Laplacian spectral radius,respectively.