On Parameters And Eccentricity Spectral Extrema Of Graphs

The spectral graph theory investigates the properties of graph by studying the eigenvalues and eigenvectors of the graph matrices(adjacency matrix,(signless)Laplacian matrix,distance matrix,etc.).In this paper,we study a new graph matrix of graph G,i.e.,the eccentricity matrixε(G).ε(G)is a graph matrix that is constructed from the distance matrix by only keeping the largest distances for each row and each column,whereas the remaining entries become null.The adjacency matrix,which is instead constructed from the distance matrix by keeping only distances equal to 1 on each row and each column.From this point of view,the adjacency matrix andε(G)are extremal among all possible distance-like matrices.In the paper,we study theε-eigenvalues and the correlation between theε-eigenvalues and the parameters of graph.The main results obtained are as follows:In Chapter 2,we characterize the graphs whoseε-spectral radius attains the minimum (resp.the second minimum value 2√2)and prove that the related graphs are determined by theirε-spectrum.In Chapter 3,we characterize the graphs that the second smallestε-eigenvalue is greater than-√15-√193,and show that all these graphs are determined by theirε-spectrum.We describe the properties of graphs satisfying the second smallestε-eigenvalue no more than-√15-√193.Moreover,we give a sufficient and necessary condition for the graphs with smallestε-eigenvalue being-2 to be determined by theirε-spectrum.In Chapter 4,by use of the mixed extension of the stars,we determine all connected graphs with exactly one positiveε-eigenvalue,and prove that these graphs are determined by theirε-spectrum.Furthermore,we identify the connected graphs with all but at most twoε-eigenvalues equal to-2 and 0.In Chapter 5,we give the extreme graphs that attain an upper bound of theε-energy of the graph and discover these graphs are strong regular graphs.We determine the HL-index of theε(G)of the graphs with exactly one positiveε-eigenvalue.