Efficient Mod Sum Labeling Scheme of Generalized Friendship Graph

Shi, Duanyin; Li, Wenyu; Zhang, Qiujie; Pan, Xiaoli

doi:10.1007/978-1-4471-4811-1_1

Cited by 1 publication

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“…However, the same name appears in connection to at least another class of graphs. The generalized friendship graph is defined by Ahmad et al [21], Arumugan and Nalliah [22], Shi et al [23] and by Jahari and Alikhani [25] as the graph consisting of r cycles of orders n 1 ≥ n 2 ≥ • • • ≥ n r having a common vertex. These graphs are also known as flowers.…”

Section: Core-satellite Graphsmentioning

confidence: 99%

Core–satellite graphs: Clustering, assortativity and spectral properties

Estrada¹,

Benzi

2017

Linear Algebra and its Applications

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Core-satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts-Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In addition, we completely describe the spectrum of the adjacency and Laplacian matrices associated with core-satellite graphs. Finally, we introduce the class of generalized core-satellite graphs and analyze their clustering, assortativity, and spectral properties. IntroductionThe availability of data about large real-world networked systems-commonly known as complex networks-has demanded the development of several graph-theoretic and algebraic methods to study the structure and dynamical properties of such usually giant graphs [1,2,3]. In a seminal paper, Watts and Strogatz [4] introduced one of such graph-theoretic indices to characterize the transitivity of relations in complex networks. The so-called clustering coefficient represents the ratio of the number of triangles in which the corresponding node takes place to the the number of potential triangles involving that node (see further for a formal definition). The clustering coefficient of a node is bounded between zero and one, with values close to zero indicating that the relative number of transitive relations involving that node is low. On the other hand, a clustering coefficient close to one indicates that this node is involved in as many transitive relations as possible. When studying complex real-world networks it is very common to report the average Watts-Strogatz (WS) clustering coefficientC as a characterization of how globally clustered a network is [1,2]. Such idea, however, was not new as reflected by the fact that Luce and Perry [5] had proposed 50 years earlier an index to account for the network transitivity, given by the total number of triangles in the graph divided by the total number of triads existing in the graph. Such index, hereafter called the graph transitivity C, was then rediscovered by Newman [6] in the context of complex networks. Here again this index is bounded between zero and one, with small values indicating poor transitivity and values close to one indicating a large one (see also [7]).It was first noticed by Bollobás [8] and then by Estrada [1] that there are certain graphs for which the two clustering coefficients diverge. That is, there are classes of graphs for which the WS average clustering coefficient tends to one while the graph transitivity tends to zero as the size of the graphs grows to infinity. The two families indentified by Bollobás [8] and by Estrada [1] are illustrated in Figure 1 and they correspond to the so-called friendship (or Dutch windmill or n-fan) graphs [9] and agave graphs [10], respectively. The friendship graphs are formed by glueing together η copies of a triangle in a common vertex. The agave graphs are for...

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Section: Core-satellite Graphsmentioning

confidence: 99%