79,979 research outputs found
Defensive alliance polynomial
We introduce a new bivariate polynomial which we call the defensive alliance
polynomial and denote it by da(G; x, y). It is a generalization of the alliance
polynomial [Carballosa et al., 2014] and the strong alliance polynomial
[Carballosa et al., 2016]. We show the relation between da(G; x, y) and the
alliance, the strong alliance and the induced connected subgraph [Tittmann et
al., 2011] polynomials. Then, we investigate information encoded in da(G; x, y)
about G. We discuss the defensive alliance polynomial for the path graphs, the
cycle graphs, the star graphs, the double star graphs, the complete graphs, the
complete bipartite graphs, the regular graphs, the wheel graphs, the open wheel
graphs, the friendship graphs, the triangular book graphs and the quadrilateral
book graphs. Also, we prove that the above classes of graphs are characterized
by its defensive alliance polynomial. A relation between induced subgraphs with
order three and both subgraphs with order three and size three and two
respectively, is proved to characterize the complete bipartite graphs. Finally,
we present the defensive alliance polynomial of the graph formed by attaching a
vertex to a complete graph. We show two pairs of graphs which are not
characterized by the alliance polynomial but characterized by the defensive
alliance polynomial
Quadratic Embedding Constants of Graph Joins
The quadratic embedding constant (QE constant) of a graph is a new
characteristic value of a graph defined through the distance matrix. We derive
formulae for the QE constants of the join of two regular graphs, double graphs
and certain lexicographic product graphs. Examples include complete bipartite
graphs, wheel graphs, friendship graphs, completely split graph, and some
graphs associated to strongly regular graphs.Comment: 20 page
Core–satellite graphs : clustering, assortativity and spectral properties
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
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