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The annihilation number a of a graph is an upper bound of the independence number α of a graph. In this article we characterize graphs with equal independence and annihilation numbers. In particular, we show that α = a if, and only if, either (1) a ≥ n 2 and α ′ = a, or (2) a < n 2 and there is a vertex v ∈ V (G) such that α ′ (G − v) = a(G), where α ′ is… (More)

Given a connected bipartite graph G, we describe a procedure which enumerates and computes all graphs H (if any) for which there is a direct product factorization G ∼ = H × K 2. 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 Brešar,… (More)

Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, α(G2H) = r(G2H) if and only if one factor is a complete graph on two vertices, and… (More)

In a classical 1986 paper by Erdös, Saks and Sós every graph of radius r has an induced path of order at least 2r − 1. This result implies that the independence number of such graphs is at least r. In this paper, we use a result of S. Fajtlowicz about radius-critical graphs to characterize graphs where the independence number is equal to the radius, for all… (More)

Let G 1 and G 2 be disjoint copies of a graph G, and let f : V (G 1) → V (G 2) be a function. Then a functigraph C(G, f) = (V, E) has the vertex set V = V (G 1) ∪ V (G 2) and the edge set E = E(G 1) ∪ E(G 2) ∪ {uv | u ∈ V (G 1), v ∈ V (G 2), v = f (u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense… (More)

The first attempt at automating mathematical conjecture-making appeared in the late-1950s. It was not until the mid-1980s though that a program produced statements of interest to research mathematicians and actually contributed to the advancement of mathematics. A central and important idea underlying this program is the Principle of the Strongest… (More)

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