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Sparse Topologies with Small Spectrum Size - Elsässer, Kralovic, Monien (2003) (Correct)
....on such graphs in a small number of steps. Key words: Laplacian of a graph; Spectra of graphs; Interconnection topologies 1 Introduction Since their introduction, spectral methods have attracted great attention and have proved to be a valuable tool for theoretical and applied graph theory [7,3]. The set of eigenvalues of the Laplace or adjacency matrix of a graph is called its Laplace or adjacency spectrum; it is one of the most important algebraic invariants of a graph. Although in general a graph is not characterized uniquely by its spectrum, there is a strong connection between its ....
....the graph: the only graph which has two distinct eigenvalues is the complete graph and its automorphism group is as rich as possible the symmetric group. Graphs having three distinct eigenvalues are called strongly regular; they have a diameter of 2 and they possess many interesting properties [17,3]. Another well studied class of highly symmetric graphs are the distance regular graphs [4] The size of their spectrum is 1 diam(G) which matches a lower bound for all graphs. We are interested in constructing graphs whose product of the vertex degree and the number of di erent eigenvalues is ....
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Biggs, N. L.:Algebraic Graph Theory, (2nd ed.), Cambridge University Press, Cambridge, 1993
The Random Walk Construction of Uniform - Spanning Trees And (Correct)
....in highly symmetric graphs For highly symmetric graphs it is reasonable to hope we can compute explicitly the degree of a vertex in the uniform random spanning tree. Proposition 9 below deals with the very special class of graphs G which satisfy the following (very strong) hypothesis. Recall [6] that G is vertex transitive if its automorphism group acts transitively on vertices. Hypothesis 7 (i) G is vertex transitive, with degree r 3. ii) For each v G and distinct neighbors w 1 , w 2 , w 3 of v, there exists a graph automorphism # such that #(v) v, #(w 1 ) w 1 , #(w 2 ) ....
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N. Biggs. Algebraic Graph Theory. Cambridge University Press, second edition, 1993.
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