We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as √ d √ d−1 .

In this paper we consider the sequence whose n th term is the number of h-vectors of length n. We show that the n th term of this sequence is bounded above by the n th Fibonacci number and bounded below by the number of integer partitions of n into distinct parts. Further we show embedded sequences that directly relate to integer partitions.

Many types of topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices are explored during past recent years. Among degree based topological indices, Zagreb indices are the oldest one and studied well. In the paper, we define a generalized multiplicative version of these indices and compute exact formulas for Polycyclic Aromatic Hydrocarbons and Jagged-Rectangle Benzenoid Systems.

Abstract.We define what it means for a Cohen-Macaulay ring to to be superstretched and show that Cohen-Macaulay rings of graded countable CohenMacaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1, n), or (1, n, 1). Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be of minimal multiplicity.

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