On Generalizations Of Strongly Regular Graphs And Related Combinatorial Structures

The concept of a distance-regular graph is an important branch of algebraic com-binatorial theory,which is closely related to finite geometry,combinatorial design and coding theory.As a distance-regular graph of diameter 2,a strongly regular graph has high reg-ularity and symmetry.The research on this graph has been a hot topic discussed by many scholars.And there are a lot of results about this graph.In recent years,how to generalize the strongly regular graph in different forms to enrich and extend the related theories of graph theory and algebraic combinatorial theory has become a hot research field and has aroused the concern of more and more experts and scholars.In this paper,we mainly study the parameters,structural characteristics,special properties and constructions of some kinds of generalizations of strongly regular graphs and obtain some innovative results.The structure of this paper is as follows:In Chapter 1,we mainly introduce some definitions of the generalizations of strongly regular graphs and construction tools of graphs.In Chapter 2,we study the connectivity of the second neighborhoods of quasi-strongly regular graphs of grade 2 and their complement and determine that the second neighborhood of any vertex of these graphs is connected except for two kinds of graphs.In Chapter 3,we study the extendability of Deza graphs and coedge-regular graphs of odd order.We show that strictly Deza graphs of odd order and strictly coedge-regular graphs of odd order are all 1（?）extendable.Furthermore,we prove that each strictly coedge-regular graph of odd order is 2（?）extendable except for the complement of C₇（the cycle of length 7）.In Chapter 4,we determine the arc-connectivity and minimum arc-cut sets of normal-ly regular digraphs and Deza digraphs.The arc-connectivity of both types of digraphs is their degree and both types of digraphs are super arc-connected except for a few digraphs.In Chapter 5,we define a generalization of quasi-strongly regular graphs from the perspective of digraphs,called directed quasi-strongly regular graphs.First,we study directed quasi-strongly regular graphs of grade 2 and obtain some parameter constraints.Moreover,using the eigenvalues of a quasi-strongly regular graph of grade 2,we determine the eigenvalues of its children.And we give some constructions of quasi-strongly regular digraphs based on Cayley digraphs,digraph product operations,weakly distance-regular digraphs and association schemes,respectively.Finally,we consider a variation of directed quasi-strongly regular graphs,called quasi-strongly regular digraphs,which is also another generalization of quasi-strongly regular graphs from the perspective of digraphs,and show some of their constructions.In Chapter 6,we extend the generalized strongly regular graphs to the category of digraphs,and propose the concept of generalized strongly regular digraphs.We obtain some parameter constraints on generalized strongly regular digraphs of grade 2 and give some construction methods of generalized strongly regular digraphs based on Cayley digraphs and digraph product operations.