By V. N. Faddeeva
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Additional info for Computational methods of linear algebra
It is known that there are no regular graphs with girth g = 5 and excess e = 1. 12 (P. K O V ~ C[SK o v ~ ] ) : Let T be an oddinteger, T 2 3, # l2 + 1 + 3, and T # l2 + 1 - 1. Then there exists no regular graph of T degree T , girth g = 5 , and excess e = 2. A lower bound for the excess has been given by N. Biggs. For a given k let the polynomials F,(z) be defined recursively by Fo(z) = 1, FI(z) = z + 1, and F,(z) = zF,-I(z) - ( k - l ) F , - 2 ( ~ ) for T 2 2. Suppose G is a graph that is regular of degree k, has girth g = 2r 1 and excess e.
G. Bridges and M. S. Shrikhande [BRSHl] use partially balanced incomplete block designs to construct graphs with six or seven distinct eigenvalues. H. D. Patterson and E. R. Williams [PAW111 measure the efficiency of a block design by taking the harmonic mean of the nonzero eigenvalues of I - A N N T where N is the block incidence matrix of the design with parameters (v, b, T , k). H. Beker and W. Haemers [BEHAI] have investigated balanced incomplete block designs with parameters (v, b, T , k, A) that have k - n for as intersection number for some of the blocks (n = T - A is the design order).
ITO [BIITl]): There is no regular graph of degree T with girth g = 6, excess e = 2 if r 5 mod 8 or r 7 mod 8. There have also been extensions of Moore graphs to the directed case. J. Bosak uses both directed and undirected edges in what he calls a partially directed graph. Such a graph is homogeneous if, for any vertex v, the number of directed edges going into v equals the number of the directed edges going out of v . A partially directed graph with r edges incident to each vertex, diameter d, and with every pair of vertices joined by a unique directed path with length less than d is cdled a ( r ,d ) Moore Graph.