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Báo cáo toán học: " Coverings, heat kernels and spanning trees"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Coverings, heat kernels and spanning trees. | Coverings heat kernels and spanning trees Fan Chung University of Pennsylvania Philadelphia Pennsylvania 19104 chung@hans.math.upenn.edu S.-T. Yau Harvard University Cambridge Massachusetts 02138 yau@math.harvard.edu Submitted May 1 1998 Accepted December 12 1998. AMS Subject Classification 05C50 05Exx 35P05 58G99. Abstract We consider a graph G and a covering G of G and we study the relations of their eigenvalues and heat kernels. We evaluate the heat kernel for an infinite k-regular tree and we examine the heat kernels for general k-regular graphs. In particular we show that a k-regular graph on n vertices has at most 2 log n i k - 1 k 1 V o 1 kn log k V k2 - 2k k 2-1 spanning trees which is best possible within a constant factor. 1 Introduction We consider a weighted undirected graph G possibly with loops which has a vertex set V V G and a weight function w V X V IB satisfying w u v w v u and w u v 0. Research supported in part by NSF Grant No. DMS 98-01446 Research supported in part by NSF Grant No. DMS 95-04834 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R12 2 If w u v 0 then we say u v is an edge and u is adjacent to v. A simple graph is the special case where all the weights are 0 or 1 and w v v 0 for all v. In this paper by a graph we mean a weighted graph unless specihed. The degree dv of a vertex v is dehned to be dv 52 w u v . A graph is regular if all its degrees are the same. For a vertex v in G the neighborhood N v of v consists of all vertices adjacent to v. This paper is organized as follows In Section 2 we dehne a covering of a graph and give several examples. In Section 3 we give the dehnitions for the Laplacian eigenvalues and the heat kernel of a graph. In Section 4 we consider the relations between the eigenvalues of a graph and the eigenvalues of its covering. In particular we give a proof for determining the eigenvalues and their multiplicities of a strongly cover-regular graph G from the eigenfunctions of the smaller graph covered by .