We know that the polyhedra corresponding to the Platonic solids are equivelar. In this article we have classified completely all the simplicial equivelar polyhedra on ≤ 11 vertices. There are exactly 27 such polyhedra. For each n ≥ −4, we have classified all the ( p, q) such that there exists an equivelar polyhedron of type { p, q} and of Euler characteristic n. We have also constructed five types of equivelar polyhedra of Euler characteristic −2m, for each m ≥ 2.

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices.In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an n-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinct n-vertex weakly regular triangulations of the torus for each n ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for each m ≥ 2. For 12 ≤ n ≤ 15, we have classified all the n-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.Proposition 1 . There are exactly 77 weakly regular combinatorial 2-manifolds on at most 15 vertices; 42 of these are orientable and 35 are non-orientable. Among these 77 combinatorial 2-manifolds, 20 are of Euler characteristic 0. These 20 are T

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic −2 must contain 12 vertices.In

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We introduce the k-stellated spheres and consider the class W k (d) of triangulated d-manifolds all whose vertex links are k-stellated, and its subclass W * k (d) consisting of the (k + 1)-neighbourly members of W k (d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W k (d) for d ≥ 2k. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of W * k (d) for d ≥ 2k + 2. As another application, we prove that, when d = 2k + 1, all members of W * k (d) are tight. We also characterize the tight members of W * k (2k + 1) in terms of their k th Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds.We also prove a lower bound theorem for triangulated manifolds in which the members of W 1 (d) provide the equality case. This generalises a result (the d = 4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W 1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight triangulated manifolds should be strongly minimal. (2010): 57Q15, 52B05. Mathematics Subject ClassificationDefinition 1.1. For 0 ≤ k ≤ d + 1, a d-dimensional simplicial complex X is said to be k-stellated if X may be obtained from S d d+2 by a finite sequence of bistellar moves, each of index < k. By convention, S d d+2 is the only 0-stellated simplicial complex of dimension d. Clearly, for 0 ≤ k ≤ l ≤ d + 1, k-stellated implies l-stellated. All k-stellated simplicial complexes are combinatorial spheres. By Pachner's theorem [25], the (d + 1)-stellated dspheres are precisely the combinatorial d-spheres.We also recall : Definition 1.2. For 0 ≤ k ≤ d + 1, a triangulated (d + 1)-dimensional ball B is said to be k-stacked if all the faces of B of codimension (at least) k + 1 lie in its boundary; i.e., if