Brendan D. McKay

Isomorph-Free Exhaustive Generation

We describe a very general technique for generating families of combinatorial objects without isomorphs. It applies to almost any class of objects for which an inductive construction process exists. In one form of our technique, no explicit isomorphism testing is required. In the other form, isomorph testing is restricted to within small subsets of the entire set of objects. A variety of different examples are presented, including the generation of graphs with some hereditary property, the generation of Latin rectangles and the generation of balanced incomplete block designs. The technique can also be used to perform unbiased statistical analysis, including approximate counting, of sets of objects too large to generate exhaustively.

The expected eigenvalue distribution of a large regular graph

Abstract Let K 1 , K 2 ,... be a sequence of regular graphs with degree v ⩾2 such that n(X i ) →∞ and ck(X i )/n(X i ) →0 as i ∞ for each k ⩾3, where n(X i ) is the order of X i , and c k (X i ) is the number of k - cycles in X 1 . We determine the limiting probability density f(x ) for the eigenvalues of X> i as i →∞. It turns out that f(x)= v 4(v−1)−v 2 2π(v 2 −x 2 ) 0 for ∦ x ∦⩽2 v-1 , otherwise It is further shown that f(x ) is the expected eigenvalue distribution for every large randomly chosen labeled regular graph with degree v .

Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2)

AbstractWe determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1/4). In particular, ifk=o(n1/2), the number of regular graphs of degreek and ordern is asymptotically

Uniform generation of random regular graphs of moderate degree

\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).

Asymptotic enumeration by degree sequence of graphs of high degree

Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastαn3g−1 for someα>0, wheregn is the number of graphs of ordern andn ≥ 7.

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Abstract We show how to generate k -regular graphs on n vertices uniformly at random in expected time O ( nk 3 ), provided k = O(n 1 3 ) . The algorithm employs a modification of a switching argument previously used to count such graphs asymptotically for k = o(n 1 3 ) . The asymptotic formula is re-derived, using the new switching argument. The method is applied also to graphs with given degree sequences, provided certain conditions are met. In particular, it applies if the maximum degree is O(∥E(G)∥ 1 4 ) . The method is also applied to bipartite graphs.

Constant time generation of free trees

Abstract A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.