和图、整和图与模和图的几个结果.pdfVIP

Journal of Mathematical Research Exposition Feb., 2008, Vol. 28, No. 1, pp. 217–222 DOI:10.3770/j.issn:1000-341X.2008.01.028 Http:// Some Results on Sum Graph, Integral Sum Graph and Mod Sum Graph 1,2 1 2 ZHANG Ming , YU Hong-quan , MU Hai-lin (1. Department of Applied Mathematics, Dalian University of Technology, Liaoning 116024, China; 2. School of Energy and Power Engineering, Dalian University of Technology, Liaoning 116024, China) (E-mail: xingche0782@) Abstract Let N denote the set of positive integers. The sum graph G+ (S) of a finite subset S ⊂ N is the graph (S, E) with uv ∈ E if and only if u + v ∈ S . A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S ⊂ N . By using the set Z of all integers instead of N , we obtain the definition of the integral sum graph. A graph G = (V, E) is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2, . . . , z − 1} so that uv ∈ E if and only if the sum, modulo z , of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill (Dm) is integral sum graph and mod sum graph, and give the sum number of Dm . Keywords sum graph; integral sum graph; mod sum graph; flower tree; Dutch m-wind-mill. Document code A MR(2000) Subject Classification 05C15 Chinese Library Classification O157.5 1. Introduction All graphs in this paper are finite and have no loops or multiple edges. We follow in general the graph-theoretic notation and terminology