Research On The α-spectra Of Graph And Related Problems

Algebraic graph theory is a branch of mathematics that studies graph related prob-lems by algebraic methods,mainly including three aspects:graph theory;The properties of graphs are studied by the method of group theory,especially the automorphism group of graphs;Objects are related with invariants such as chromatic number and matching number of graphs.Spectral theory mainly studies the relationship between the properties of graphs and the characteristic polynomials,eigenvalues and eigenvectors of graph correlation matrices(such as adjacency matrix,Laplace matrix and distance matrix).The main methods come from linear algebra and matrix theory.In 2017,the concept of A_αmatrix of a graph G was put forward by Nikiforov,which is hybrid of A(G)and D(G)similar to the signless Laplacian matrix,where A(G),D(G)are the adjacency matrix and degree matrix of graph G,respectively.The study of A_αmatrix has started attracting attention of researchers recently.This paper consists of eight chapters.It mainly studies the A_αmatrix andα-spectral radius of simple graph,hypergraph,directed(hyper)graph and weighted graph,respectively.This paper is organizeded as follows:·In Chapter 1 and 2,we mainly introduces the background,the research status,the concepts and lemmas of spectral theory and graphs.·In Chapter 3,we characterize the extremal digraphs with the maximal or minimal α-spectral radius among some digraph classes such as rose digraphs,generalizedΘ-digraphs and tri-ring digraphs with given size m.As a by-product of our main results,an open problem in[The signless Laplacian spectral radius of some strongly connected digraphs,Indian J.Pure Appl.Math.49(1)(2018)113–127]is answered.Furthermore,we determine the digraphs with the first three minimalα-spectral radius among all strongly connected digraphs.Meanwhile,we determine the unique digraph with the fourth minimalα-spectral radius among all strongly connected digraphs for 0≤α≤1/2.·In Chapter 4,we discuss the methods for comparingα-spectral radius of graphs.As applications,we characterize the graphs with the maximalα-spectral radius among all unicyclic and bicyclic graphs of order n with diameter d,respectively.Finally,we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order n with diameter d.From our conclusion,it is known that the result of Pai and Liu in[On the signless Laplacian spectral radius of bicyclic graphs with fixed diameter.Ars Combinatoria 2017,249–265]is wrong.·In Chapter 5,upper bounds on theα-spectral radius of an irregular weighted graph are given in terms of the number of vertices,diameter and maximum vertex weight.At the same time,an upper bounds on theα-spectral radius of a k-connected irregular weighted graph is also established in terms of the number of vertices,connectivity,maximum vertex weight,minimum edge weight and sum of vertex weights.·In Chapter 6,we show how theα-spectral radius changes under the edge grafting operations on connected k-uniform hypergraphs.We characterize the extremal hypertree forα-spectral radius among k-uniform non-caterpillar hypergraphs with given order,size and diameter.We also characterize the supertree with the second largestα-spectral radius among all k-uniform supertrees on n vertices.·In Chapter 7,we show the lower bound of the ratio of the largest component to the smallest component in theα-Perron vector of hypergraph.By using the technique of the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor,we obtain some new bounds by applying these bounds to a strongly connected k-uniform(directed)hypergraph.·In Chapter 8,we summarize the paper and pose some questions and conjections.