 Journal of Software Engineering and Applications, 2013, 6, 29-32 doi:10.4236/jsea.2013.63b007 Published Online March 2013 (http://www.scirp.org/journal/jsea) Copyright © 2013 SciRes. JSEA 29Felicitous Labellings of Some Network Models* Jiajuan Zhang1, Bing Yao1, Zhiqian Wang2, Hongyu Wang1, Chao Yang1, Sihua Yang1 1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China; 2School of Mathematics Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China. Email: yy}918@163.com Received 2013 ABSTRACT Building up graph models to simulate scale-free networks is an important method since graphs have been used in re-searching scale-free networks. One use labelled graphs for distinguishing objects of communication and information networks. In this paper some methods are given for constructing larger felicitous graphs from smaller graphs having special felicitous labelling s , and some network models are shown to be felicitous. Keywords: Felicitous Labelling; Set-Ordered Felicitous Labelling; Symmetric Graphs ;Trees 1. Introduction Graphs have been used in researching scale-free net-works (,). Many graph models can be labelled for distinguishing objects in communication networks. Ichishma and Oshima  investigated the relationship between partitional graphs and strongly graceful graphs and partitional graph s and stron gly felicitou s graphs . Yao, Chen, Yao and Cheng  show several relationships among several well-known labellings including felicitous labelling. In , among the graphs known to be felicitous are: Cn except when n2 (mod 4); Km,n when m,n>1; P2C2n+1; P2C2n; P3C2n+1; SmC2n+1; Kn if and only if n4; Pn+Km; the friendship graph (3)nC for n odd; PnC3; PnCn+3. It has been noticed that some felicitous graphs in literature can be constructed, such as some classes of felicitous trees are obtained in . Graham and Sloane  conjectured: Every tree is felicitous. We will present several methods for constructing larger felicitous graphs from smaller graphs having special felicitous la-bellings, and show some network models to be felicitous, such as edge-symmetric and near-symmetric graphs that are related with some models of self-similar and hierar-chical networks in current research of complex networks. Standard terminology and notation of graph theory are used here. Simple graphs are finite, undirected, no multi-ple edges and loopless, unless otherwise specified. The shorthand notation [m,n] stands for an integer set {m, m+1,…, n} with n>m0. A (p,q)-graph is a simple graph with p vertices and q edges. A proper labelling f of a (p,q)-graph G is a mapping from V(G) to [m,n] such that f(x)f(y) for distinct x,y V(G). Definition 1  Suppose that a (p,q)-graph G has a proper labelling f: V(G)[0,q]. The edge label f(uv) of each edge uvE(G) is defined as f(uv)=f(u)+f(v) (mod q). If the edge label set {f(uv): uvE(G)}=[0,q1], then we say both G and f to be felicitous. For the purpose of simplicity, we write f(S)={f(x): xSV(G)} and f(E(G))={f(uv): uvE(G)}={f(u)+f(v) (mod q): uvE(G)} and fG={f(u)+f(v): uvE(G)} for a felicitous labelling f of a (p,q)-graph G throughout this paper. Very often, the labelling h(x)=qf(x) for each verte xV(G) is called the dual labelling of the labelling f. The notation S (mod q) stands for the set {x (mod q): xS} for a set S of non-negative integers. For a set [6,14]={6,7,8,9,10,11,12,13,14}, as an example, [6,14] (mod 10) ={0 ,1,2 ,3,4 ,6,7 ,8,9 }, and [6 ,14] (bmod 7) =[0, 7]; for S={3,7,9,10,12,13}, then S (mod 9)={0,1,3,4,7}. In  and , the authors introduced the set-ordered graceful labellings and the set-ordered odd-graceful la-bellings: Let (X,Y) be the bipartition of a bipartite (p,q)-graph G. If G admits a (an odd-)graceful labelling f such that max{f(u):u X}q, so g(x)+g(y)=2q[f(x)+f(y)]=2q[f(x)+f(y)q+q]=qf(xy). Therefore, g(xy)=g(x)+g(y) (mod q)=qf(xy) for xyE(G) and f(xy)0. If G is bipartite, let (X,Y) be the bipartition of G. Suppose that f is set-ordered felicitous, that is, f(X) 0 and g(yt)