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Báo cáo toán học: " Constructing fifteen infinite classes of nonregular bipartite integral graphs"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Constructing fifteen infinite classes of nonregular bipartite integral graphs. | Constructing fifteen infinite classes of nonregular bipartite integral graphs Ligong Wang1 yand Cornells Hoede2 1 Department of Applied Mathematics School of Science Northwestern Polytechnical University Xi an Shaanxi 710072 P. R. China. ligongwangnpu@yahoo.com.cn 2Department of Applied Mathematics Faculty of Electrical Engineering Mathematics and Computer Science University of Twente P.O. Box 217 7500AE Enschede The Netherlands. hoede@math.utwente.nl Submitted Oct 5 2007 Accepted Dec 16 2007 Published Jan 1 2008 Mathematics Subject Classifications 05C50 11D09 11D41 Abstract A graph is called integral if all its eigenvalues of the adjacency matrix are integers. In this paper the graphs S1 t K1 t S2 n t S3 m n t S4 m n p q S5 m n S6 m n t S8 m n S9 m n p q S10 n S13 m n S17 m n p q S18 n p q t S19 m n p t S20 n p q and S21 m t are defined. We construct the fifteen classes of larger graphs from the known 15 smaller integral graphs S1 S6 S8 S10 S13 S17 S21 see also Figures 4 and 5 Balinska and Simic Discrete Math. 236 2001 13-24 . These classes consist of nonregular and bipartite graphs. Their spectra and characteristic polynomials are obtained from matrix theory. We obtain their integral property by using number theory and computer search. All these classes are infinite. They are different from those in the literature. It is proved that the problem of finding such integral graphs is equivalent to solving Diophantine equations. We believe that it is useful for constructing other integral graphs. The discovery of these integral graphs is a new contribution to the search of integral graphs. Finally we propose several open problems for further study. 1 Introduction We use G to denote a simple graph with vertex set V G v1 v2 . vng and edge set E G . The adjacency matrix A A G ữịj of G is an n X n symmetric matrix of 0 s Supported by National Science Foundation of China and Natural Science Basic Research Plan in Shaanxi Province of China. y Corresponding author. THE .