Core–satellite graphs: Clustering, assortativity and spectral properties

Abstract

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...

Loss of structural balance in stock markets

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Integer Laplacian eigenvalues of chordal graphs

“…We establish the relation between classic invariants of graphs and their integer Laplacian eigenvalues, pointing out how their computation can be efficiently implemented. We focus on a subclass of chordal graphs [14], the block duplicate graphs, introduced by Golumbic and Peled [10] and also defined as strictly chordal graphs based on hypergraph properties [15,16]; this class contains the classes of block graphs [14], block-indifference graphs [2], the generalized coresatellite graphs [6] and the (k, t)-split graphs [3]. In Section 2, we review results concerning general graphs, showing that the number of universal vertices and the degree of false and true twins can provide integer Laplacian eigenvalues and their multiplicities.…”

On the Laplacian Spectra of Some Double Join Operations of Graphs

“…Determining the spectra of many graph operations is a basic and very meaningful work in spectral graph theory. Up till now, many graph operations such as Cartesian product, Kronecker product, graph with k (edge)-pockets, corona, edge corona, some variants of (edge)corona, join, some variants of join have been introduced and the adjacency spectra (Laplacian spectra, signless Laplacian spectra as well) of these graph operations have also been determined in terms of the corresponding spectra of the factor graphs in [1,2,3,4,6,8,12,14,15,16,20,21]. Moreover, it is known that the corresponding spectra of these graph operations can be used to construct infinitely many pairs of cospectral graphs [1,3,4,8,11,16,20], infinitely families of integral graphs [2,15] and to investigate many other properties of graphs, such as the Kirchhoff index [16,17,21], the number of spanning trees [4,14,16] and so on.…”

“…Cardoso et al [6] characterized adjacency and Laplacian spectra of the H-join operation of graphs. Estrada and Benzi [12] discussed the clustering, assortativity and spectral properties of core-satellite graphs. Remark that the core-satellite graph named in [12] is a special join of some complete graphs.…”

“…Estrada and Benzi [12] discussed the clustering, assortativity and spectral properties of core-satellite graphs. Remark that the core-satellite graph named in [12] is a special join of some complete graphs. In [15], the adjacency spectra of the subdivision vertex(edge) joins of graphs were computed in terms of the corresponding spectra of two regular graphs.…”