Graphs. the degree of a vertex

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  • This is an electronic version of the second (2000) edition of the above Springer book, from their series Graduate Texts in Mathematics, vol. 173. The cross-references in the text and in the margins are active links: click on them to be taken to the appropriate page. The printed edition of this book can be ordered from your bookseller, or electronically from Springer through the Web sites referred to below.

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  • We propose two alternative measures of the local irregularity of a graph in terms of its vertex degrees and relate these measures to the order and the global irregularity of the graph measured by the difference of its maximum and minimum vertex degree.All graphs will be simple and finite. Let G = (V,E) be a graph of order n = |V |. The degree and the neighbourhood of a vertex u 2 V will be denoted by d(u) and N(u). The maximum and minimum degree of G will be denoted by (G) and (G).

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  • The betweenness centrality (BWC) of a vertex is a measure of the fraction of shortest paths between any two vertices going through the vertex and is one of the widely used shortest path-based centrality metrics for the complex network analysis.

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  • Classification of Mechanisms Using graph representation, mechanism structures can be conveniently represented by graphs. The classification problem can be transformed into an enumeration of nonisomorphic graphs for a prescribed number of degrees of freedom, number of loops, number of vertices, and number of edges. The degrees of freedom of a mechanism are governed by Equation (4.3). The number of loops, number of links, and number of joints in a mechanism are related by Euler’s equation, Equation (4.5). The loop mobility criterion is given by Equation (4.7).

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