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The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Traceminimal graphs and Doptimal weighing designs, preprint
"... Let G(v, δ) be the set of all δregular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called traceminimality. In particular, all strongly regular graphs with no triangles and some cages are traceminimal. These graphs play an important r ..."
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Let G(v, δ) be the set of all δregular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called traceminimality. In particular, all strongly regular graphs with no triangles and some cages are traceminimal. These graphs play an important role in the statistical theory of Doptimal weighing designs. Each weighing design can be associated with a (0, 1)matrix. Let Mm,n(0, 1) denote the set of all m × n (0,1)matrices and let G(m, n) = max{det X T X: X ∈ Mm,n(0, 1)}. A matrix X ∈ Mm,n(0, 1) is a Doptimal design matrix if det X T X = G(m, n). In this paper we exhibit some new formulas for G(m, n) where n ≡ −1 (mod 4) and m is sufficiently large. These formulas depend on the congruence class of m (mod n). More precisely, let m = nt + r where 0 ≤ r < n. For each pair n, r, there is a polynomial P (n, r, t) of degree n in t, which depends only on n, r, such that G(nt + r, n) = P (n, r, t) for all sufficiently large t. The polynomial P (n, r, t) is computed from the characteristic polynomial of the adjacency matrix of a traceregular graph whose degree of regularity and number of vertices depend only on n and r. We obtain explicit expressions for the polynomial P (n, r, t) for many pairs n, r. In particular we obtain formulas for G(nt + r, n) for n = 19, 23, and 27, all 0 ≤ r < n, and all sufficiently large t. And we obtain families of formulas for P (n, r, t) from families of traceminimal graphs including bipartite graphs obtained from finite projective planes, generalized quadrilaterals, and generalized hexagons. Keywords: Doptimal weighing design, traceminimal graph, regular graph, strongly regular graph, girth, cages, generalized polygons AMS Subject Classification:
Decomposition Characterizations of Classes of 2Connected Graphs
"... By applying the Tutte decomposition of 2connected graphs into 3block trees we provide unique structural characterizations of several classes of 2connected graphs, including minimally 2connected graphs, minimally 2edgeconnected graphs, critically 2connected graphs, critically 2edge ..."
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By applying the Tutte decomposition of 2connected graphs into 3block trees we provide unique structural characterizations of several classes of 2connected graphs, including minimally 2connected graphs, minimally 2edgeconnected graphs, critically 2connected graphs, critically 2edgeconnected graphs, 3edgeconnected graphs, 2connected cubic graphs and 3connected cubic graphs. We also give a characterization of minimally 3connected graphs. 1