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For a graph G, let 9(G) be the family of strong orientations of G, d(G) = min{d(D) / D t 9' (G)} and p(G) = d(G)-d(G), where d(G) and d(D) are the diameters of G and D respectively. In this paper we show that p(G) = 0 if G is a Cartesian product of (I) paths, and (2) paths and cycles, which satisfy some mild conditions.

In this paper, we present a formula for computing the Tutte polynomial of the signed graph formed from a labeled graph by edge replacements in terms of the chain polynomial of the labeled graph. Then we define a family of 'ring of tangles' links and consider zeros of their Jones polynomials. By applying the formula obtained, Beraha-Kahane-Weiss's theorem… (More)

Note On a conjecture concerning optimal orientations of the cartesian product of a triangle and an odd cycle Abstract Let G × H denote the cartesian product of the graphs G and H , and Cn the cycle of order n. We prove the conjecture of Konig et al. that for n ¿ 2, the minimum diameter of any orientation of the graph C3 × C2n+1 is n + 3.

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and µ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that µ(D(G)) ≤ n(G) + 1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with µ(D(G)) = n(G) + 1.… (More)