Constructing MDS Matrices And Studying Double Graphs

Circulant MDS matrices and double graphs have always been research hotspots.This thesis main research on a construction of circulant MDS matrices and expected hitting time and related invariants of double graphs.On the one hand,we study a construction of circulant MDS matrices.Circulant MDS matrices are not only important in coding theory but also have many applications in cryptography.Constructing circulant MDS matrices is thus of significance in theory and practice.In this thesis,a new construction of circulant MDS matrice over finite fields is presented.As a consequence,a class of circulant involutory MDS matrices over finite fields of odd characteristic are constructed.On the other hand,we consider expected hitting time and related invariants of double graphs.Let G be a simple connected graph and let D_Gbe its double graph.First,a relation for the expected hitting time between any two vertices of D_Gand G is displayed.As a consequence,a closed-form formula for the resistance distances of any two vertices of D_Gin terms of those of G can be derived,which is a main result obtained in[J.Appl.Math.Comput.50(2016)1-14].A closed-form formula of cover cost(resp.reverse cover cost)for any vertex of D_Gis also determined.As a consequence,the unique double tree with the minimum cover cost is identified and the sharp lower and upper bounds of the reverse cover cost together with their corresponding extremal graphs for double trees and double unicyclic graphs are characterized as well.Finally,the degree-Kirchhoff index,the Kemeny's constant and the number of spanning trees of D_Gare also studied.