The largest known 3-regular 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 3-regular, diameter-3 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...

Let G(v, δ) be the set of all δ-regular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called trace-minimality. In particular, all strongly regular graphs with no triangles and some cages are trace-minimal. These graphs play an important role in the statistical theory of D-optimal 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 D-optimal 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 trace-regular 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 trace-minimal graphs including bipartite graphs obtained from finite projective planes, generalized quadrilaterals, and generalized hexagons. Keywords: D-optimal weighing design, trace-minimal graph, regular graph, strongly regular graph, girth, cages, generalized polygons AMS Subject Classification:

By applying the Tutte decomposition of 2--connected graphs into 3--block trees we provide unique structural characterizations of several classes of 2--connected graphs, including minimally 2--connected graphs, minimally 2--edge--connected graphs, critically 2--connected graphs, critically 2--edge--connected graphs, 3--edge--connected graphs, 2--connected cubic graphs and 3--connected cubic graphs. We also give a characterization of minimally 3connected graphs. 1