Graphs and digraphs

To conclude this section, we note quickly a few other asymptotic enumerations. In [PalE70] Palmer found an asymptotic estimate for the number of self-complementary graphs and digraphs. Robinson [RobR76] and Stanley [StaR73] enumerated acyclic digraphs, and obtained the asymptotic estimate... 

HarF67a Harary, F. Enumeration of graphs and digraphs. A seminar in graph theory. (F. Harary, ed.) Holt, Rinehart and Winston, New York (1967) 34-41. 

OttR48 Otter, R. The number of trees. Ann. Math. 49 (1948) 583. PalE70 Palmer, E. M Asymptotic formulas for the number of self-complementary graphs and digraphs. Mathematika 17... 

ReaR63 Read, R. C. On the number of self-complementary graphs and digraphs. J. London Math. Soc. 38 (1963) 99-104. ReaR68 Read, R. C. The use of -functions in combinatorial analysis. Canad. J. Math. 20 (1968) 808-841. 

A. Description and Characterization of Flow Networks 1. Graphs and Digraphs... 

In order to formulate a distance matrix for unconnected graphs and digraphs, the distance between two vertices which are not connected by a wdk is defined as oo (in infinite simple graphs geodesics of infinite length may also occur). Hence, the distance matrix of a graph composed of two components, G Q U will exhibit the following block form ... 

Chartrand G, Lesniak L (1986) Graphs and digraphs. Wadsworth, Behnont... 

Abamowitz, M., Stegun, I.A., editors. (1965). Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York. Bang-Jensen, J., Gutin, G. (2002). Digraphs Theory, Algorithms and Applications. Springer, New York. 

A (finite) directed graph or digraph consists of a finite set of vertices and a set of ordered pairs of vertices called arcs. We denote by VG and Eg the set of vertices and arcs of the digraph G, respectively. Given an ordering of the vertices, the adjacency matrix of a digraph G on n vertices, denoted by AG, is the (0, l)-matrix where the ij-th element... 

For each bond, it is useful to define a positive sense in the direction of the unit with higher or equal index, which will be coincident for the Nrj reactions considered above. So, there is a total number of Nz = 2Nr — Nrs kinds of directed bonds Zi, incident on monomer units X. -, the set of monomer units and bonds are the vertices and edges of a directed graph (or digraph). Vector z, of size Nz, therefore contains the indices of the repeating units to which each directed bond points. 

In the foregoing discussion the properties of the incidence matrix and the cycle matrix were illustrated in terms of a cyclic digraph, but the results on the ranks of these matrices actually hold true for any connected digraph with N vertices. For an undirected graph, M and C contain only 0 and 1 (sometimes referred to as binary matrices), mathematical relations of identical form are obtained except that modulo 2 arithmetic2 is used instead of ordinary arithmetic. The ranks of M and defined in terms of modulo 2 arithmetic are JV — 1 and C, as before, and Eqs. (10) and (11) are modified to read... 

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