Research On Energy And Spectrum Problems Of Degree-distance Weighted Matrix And Laplacian Matrix Of Graphs

In molecular chemistry,there is a popular method to construct a graph model,where the vertices represent atoms,and the edges represent chemical bonds between atoms;call it molecular graph.For a molecular graph G,assume f(x,y)is a symmetric function,with every vertex pair u,v∈V(G)weighted by f(u,v),(?)f(u,v)is named after a chemical index of G.On the other hand,denote the degree of vertex v by dv,if we render f(x,y)to be a weight function of the adjacency matrix or(signless)Laplacian matrix of G,with the(i,j)-entry weighted by f(di,dj)or-f(di,dj),the resulting matrix is called a degree-weighted matrix or a degree-weighted(signless)Laplacian matrix.Based on molecular graph model,the classical graph energy problem comes from Huckel Molecular Orbital Theory(HMO),from which,it is proved that the total πelectron energy in a conjugated hydrocarbon is given by the sum of singular values of the unweighted adjacency matrix corresponding to its molecular graph.In 1978,Gutman introduced it into graph theory as a mathematical concept.As a supplement of graph energy,Gutman and Zhou first introduced the concept of Laplacian energy;later in 2008,Liu and Liu proposed the concepts of signless Laplacian energy and Laplacianenergy like invariant;on the basis of their work,in 2009,Jooyandeh et al.brought up the concept of incidence energy,extending the definition of Laplacian-energy like invariant to signless Laplacian matrices.Thereafter,based on different chemical practices,various forms of graph energy have been invented.On the whole,these energies are expressed with a sum of singular values of some matrix,which is no longer the classical adjacency matrix nor(signless)Laplacian matrix,but a matrix whose entries are weighted by a symmetric function.Usually there are two cases:First,it is a degree-weighted matrix with some f(x,y);second,denote the distance of two vertices u and v by D(u,v),it is a matrix weighted by some function on D(u,v),whose(u,v)-entry equals f(D(u,v)).In 1994,Dobrynin and Kochetova put forward a new type of topological index determined by the values of both distances and degrees of vertices,and recently this new type of indices become more and more popular.From this kind of indices,it is natural to define a new kind of matrices with mixed degree-distance-based entries,since a 2-dimensional matrix contains much more data than a single index,and its algebraic property will show more structural information of a molecular.We name it after degree-distance-weighted matrix.On the basis of the new concept,we defined degree-distance-weighted(signless)Laplacian matrix,and managed to generalize(signless)Laplacian energy,Laplacianenergy like invariant and incidence energy to degree-distance-weighted(signless)Laplacian matrices.This thesis focuses on the variant energies of degree-distance-weighted adj acency matrices and(signless)Laplacian matrices,and extremal spectral radius problem.The results are in the following:1.Within the Erd(?)s-Rényi random graph model Gn,p,we calculated the asymptotic value of energy of degree-distance-weighted adjacency matrices of random graphs,with weight function f(x,y);what is more,we listed all the existing chemical index functions,and calculated the exact asymptotic values of graph energy under them.This result generalizes the work on random graph energy of unweighted adjacency matrices in LAA 435(2011)by Du et al.,and the work on random graph energy of degreeweighted adjacency matrices in DAM 284(2020)by Li et al.2.Within the Erd(?)s-Rényi random graph model Gn,p,we studied the asymptotic values of(signless)Laplacian energy,Laplacian-energy like invariant and incidence energy of degree-distance-weighted(signless)Laplacian matrices.These results generalize the work on these energies of unweighted(signless)Laplacian matrices in JMAA 368(2010)and MATCH 64(2010))by Du et al.3.We found the extremal trees with the largest and smallest spectral radius of degree-weighted adj acency matrices with weight function f(di,dj).We also described the structures of extremal trees.This result generalizes the work on extremal trees of Randic matrices and ABC matrices by Cruz,Monsalve,and Rada et al.What is more,it is the first attempt to unify the spectral study of weighted adjacency matrices of graphs weighted by some topological indices.4.We confirmed the validity of strong Akbari’s Conjecture for threshold graphs,that is,the energy of threshold graphs is more than the sum of average degree and the order of graph.This enriches the work of the validity of strong Akbari’s Conjecture for bipartite and planar graphs.