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We define the excess degree ξ(P ) of a d-polytope P as 2f 1 − df 0 , where f 0 and f 1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple.It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d − 2, and the value d − 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even.The excess degree is then applied in three different settings. It is used to show that polytopes with small excess (i.e. ξ(P ) < d) have a very particular structure: provided d = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Secondly, we characterise completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). And thirdly all pairs (f 0 , f 1 ), for which there exists a 5-polytope with f 0 vertices and f 1 edges, are determined.sum of two polytopes Q + R is defined to be {x + y : x ∈ Q, y ∈ R}, and a polytope P is said to be homothetic to a polytope Q if Q = λP + t for λ > 0 and t ∈ R d . Polytopes with excess d − 2 exist in all dimensions, but their structure is quite restricted: either there is a unique nonsimple vertex, or there is a (d − 3)-face containing only vertices with excess degree one. Polytopes with excess d − 1 are in one sense even more restricted: they can exist only if d = 3 or 5. However a 5-polytope with excess 4 may also contain vertices with excess degree 2. On the other hand, d-polytopes with excess degree d or d + 2 are exceedingly numerous.In §5, we characterise all the decomposable d-polytopes with 2d + 1 or fewer vertices; this incidentally proves that a conditionally decomposable d-polytope must have at least 2d + 2 vertices.The final application, in §7, is the completion of the (f 0 , f 1 ) table for d ≤ 5; that is, we give all the possible values of (f 0 , f 1 ) for which there exists a d-polytope with d = 5, f 0 vertices and f 1 edges. The solution of this problem for d ≤ 4 was already well known [5, Chap. 10]. The same result has recently been independently obtained by Kusunoki and Murai [13]. Our proof requires some results of independent interest; in particular, a characterisation of the 4-polytopes with 10 vertices and minimum number of edges (namely, 21); this is completed in §6. We have more comprehensive results characterising polytopes with a given number of vertices and minimum possible number of edges, details of which will appear elsewhere [18].Most of our results tacitly assume that the dimension d is at least 3. When d = 2, all polytopes are both simple and simplicial, and the reader can easily see which theorems remain valid...

We consider bipartite graphs of degree DZ2, diameter D 5 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (D, 3, À2) -graphs. We prove the uniqueness of the known bipartite (3, 3, À2) -graph and bipartite (4, 3, À2)-graph. We also prove several necessary conditions for the existence of bipartite (D, 3, À2)graphs. The most general of these conditions is that either D or DÀ2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when D 5 6 and D 5 9, we prove the non-existence of the corresponding bipartite (D, 3, À2)-graphs, thus establishing that there are no bipartite (D, 3, À2)-graphs, for 5rDr10.Moreover, for D odd, by using a more careful counting argument, we obtain Proposition 1.2.Proposition 1.2. For do2½ðD À 1Þ þ ðD À 1Þ 3 þ Á Á Á þ ðD À 1Þ DÀ2 , DZ3 and odd DZ3, a bipartite (D, D, Àd)-graph is regular.

In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D). Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called Moore graphs, turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is, ({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect. Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1, ({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is 2D. Second, and most important, we prove the non-existence of ({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome, together with a proof on the non-existence of (4, 3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second census of ({\Delta},D,-2)-graphs known at present, the first being the one of (3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other results of this paper include necessary conditions for the existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5 such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure