Some Analysis On Simplicial Complexes

In this paper,we study heat kernels of Hodge-Laplacian,quantum walks and signless 1-Laplacian on simplicial complexes.In 1945,B.Eckmann generalized the graph Laplacian to simplicial complexes and proved the discrete version of the Hodge theorem.In this paper,we prove the Davies-Gaffney-Grigor'yan Lemma for discrete Hodge-Laplacian on simplicial complexes.Quantum walks are becoming more and more important in discrete geometry analysis and spectral theory and are also advanced tools for building quantum al-gorithms.According to the study of N.Konno,the behavior of quantum random walks is striking different from that of classical random walks.In Konno distri-bution,it is obvious to see the peculiar phenomena on the interval[-1,1].So quantum walks are widely used in various fields.In this paper,we give a new defi-nition of quantum walks on simplicial complexes according to the spirit of topology algebra.Moreover,this definition has some relation with Hodge-Laplacian on sim-plicial complexes.We prove the relation between the spectrum of corresponding discriminant operators and orientability and combinatorial structure of simplicial complexes mathematically.And we also discuss the finding probability and sta-tionary measures.Recently,the graph 1-Laplacian has been developed by the work of M.Hein,T.Buhler and K.C.Chang.In this paper,we study the relation between eigen-values of signless 1-Laplacian and combinatorial structure of simplicial complexes and obtain some non-trivial results.Moreover,we describe the constructions and their effect on the eigenvalues.