Craig E. Larson

Graph-theoretic independence as a predictor of fullerene stability

The independence number of the graph of a fullerene, the size of the largest set of vertices such that no two are adjacent (corresponding to the largest set of atoms of the molecule, no pair of which are bonded), appears to be a useful selector in identifying stable fullerene isomers. The experimentally characterized isomers with 60, 70 and 76 atoms uniquely minimize this number among the classes of possible structures with, respectively, 60, 70 and 76 atoms. Other experimentally characterized isomers also rank extremely low with respect to this invariant. These findings were initiated by a conjecture of the computer program Graffiti.

Direct Product Factorization of Bipartite Graphs with Bipartition-reversing Involutions

Given a connected bipartite graph

DOMINATION IN FUNCTIGRAPHS

G

A parallel algorithm for computing the critical independence number and related sets

, we describe a procedure which enumerates and computes all graphs

NOTES ON THE INDEPENDENCE NUMBER IN THE CARTESIAN PRODUCT OF GRAPHS

H

Automated conjecturing I

(if any) for which there is a direct product factorization

Construction of planar 4-connected triangulations

G\cong H\times K_2

A Characterization of Graphs Where the Independence Number Equals the Radius

. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Bresar, Imrich, Klavzar, Rall, and Zmazek [Finite and infinite hypercubes as direct products, Australas. J. Combin., 36 (2006), pp. 83-90, and Hypercubes as direct products, SIAM J. Discrete Math., 18 (2005), pp. 778-786].

Automated Conjecturing I: Fajtlowicz's Dalmatian Heuristic Revisited (Extended Abstract)

Let G1 and G2 be disjoint copies of a graph G, and let f : V (G1) → V (G2) be a function. Then a functigraph C(G,f) = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | u ∈ V (G1),v ∈ V (G2),v = f(u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let γ(G) denote the domination number of G. It is readily seen that γ(G) ≤ γ(C(G,f)) ≤ 2γ(G). We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.

Automated conjecturing III: Property-relations conjectures

An independent set I c is a critical independent set if ∣ I c ∣ − ∣ N ( I c )∣ ≥ ∣ J ∣ − ∣ N ( J )∣ , for any independent set J . The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. The existing algorithm runs in O ( n 2. 5 √( m /log n )) time for a graph G with n = ∣ V ( G )∣ vertices and m edges. It is demonstrated here that there is a parallel algorithm using n processors that runs in O ( n 1. 5 √( m /log n )) time. The new algorithm returns the union of all maximum critical independent sets. The graph induced on this set is a Konig-Egervary graph whose components are either isolated vertices or which have perfect matchings.