Edge-grafting Theorems On The Distance Spectral Radius Of Graphs And Their Applications

The distance matrix D(G)=(d(u,v))u,u∈VG is a symmetric real-matrix with real eigenvalues. The largest eigenvalue of D(G) is called the distance spectral radius, and is denoted by ρ(G). In this paper, four edge-grafting transformations were introduced to study the mathematical properties of the distance spectral radius of graphs. Using these nice properties, the graphs with minimal distance spectral radius among all the n-vertex unicyclic graphs with given matching number are characterized, which also minimizes the distance graph energy in the same class of graphs.In this paper, we determined the graph with minimal distance spectral radius among all n-vertex unicyclic graphs with given matching number m, which also minimizes the distance graph energy in the same class of graphs.This paper which stands on the basis of previous results, do further research on the distance spectral radius of all the n-vertex unicyclic graphs with given matching number. The concrete content is in the following:·In the first two chapters, we introduce the background and significance of the research, including the development of a representative at home and abroad regarding this aspect. Based on this research background and profound dis-cussion, by using deep-going analysis, it fully shows the main work’s necessity and innovation. Then we give some basic definitions and symbols about this paper.·In Chapter3, we give some edge-grafting theorems.·In Chapter4, we give the applications of the edge-grafting theorems and the main conclusion of the article.·In Chapter5, we give the summarize and the conjecture of this article.