Positive, Negative Inertia Index And Nullity Of Two Kinds Of Tricyclic Graphs And Some Properties Of One-step Fuzzy Graph

Abstract Using the method of combining graph theory and fuzzy mathemat-ics and Matlab software, a method of calculating the positive, negative inertia index and nullity of two kinds of tricyclic graphs is given; properties and decomposition theorem of a special fuzzy graph(i.e. one-step fuzzy graph) will be studied.The arrangement of this paper is as follows:Chapter one:Preliminaries. In this chapter, we give the basic concepts and results of graph, fuzzy graph and their operations.Chapter two:Positive, negative inertia index and nullity of two kinds of tri-cyclic graphs. Firstly, the concepts of α-graph、β-graph、core and k-connection graph are introduced. By means of deleting pendant trees and compressing inter-nal paths, a method of calculating the positive, negative inertia index and nullity of the two special kinds of tricyclic graphs is given:the positive, negative inertia index and nullity of one-type tricyclic graphs equal to the sum of those of some trees and bicyclic graphs; the positive, negative inertia index and nullity of two-type tricyclic graphs equal to the sum of those of some trees and simple tricyclic graphs, the positive, negative inertia index and nullity of these simple tricyclic graphs can be calculated by Matlab. Finally, for the two kinds of tricyclic graphs a conjecture about difference of the positive and negative inertia index of a graph proposed by predecessors is verified.Chapter three:Properties of one-step fuzzy graph. Firstly, the notions of one-step fuzzy graph, Hamiltonian fuzzy graph, r-regular fuzzy graph, bipartite fuzzy graph and connected fuzzy graph are defined. Structures of strong one-step Hamil-tonian fuzzy graphs are obtained, etc. Secondly, simpler expressions of operations of cartesian product, composition, and complete of strong one-step fuzzy graphs, and some properties of strong one-step fuzzy graphs which preserved under carte-sian product are given. Finally, decomposition theorem of one-step fuzzy graphs is discussed and it is proved that every fuzzy graph can be decomposed into one-step fuzzy graphs.