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The design spectrum of a simple graph $G$ is the set of positive integers $n$ such that there exists an edgewise decomposition of the complete graph $K_n$ into $n(n - 1)/(2 |E(G)|)$ copies of $G$. We compute the design spectra for 7788 6-regular graphs with 12 vertices., Comment: 261 pages. The paper without the appendix will be submitted to a journal

A generalized pentagonal geometry PENT($k$,$r$,$w$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point, $x$, the set of points not collinear with $x$ forms the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry. If $w = k$, the structure is called a pentagonal geometry and denoted by PENT($k$,$r$). The deficiency graph of a PENT($k$,$r$,$w$) has as its vertices the points of the geometry, and there is an edge between $x$ and $y$ precisely when $x$ and $y$ are not collinear. Our primary objective is to investigate generalized pentagonal geometries PENT($k$,$r$,$w$) where the deficiency graph has girth 4. We describe some construction methods, including a procedure that preserves deficiency graph connectedness, and we prove a number of theorems regarding the existence spectra for $k = 3$ and various values of $w$. In addition, we present some new PENT(4,$r$) (including PENT(4,25)) and PENT(5,$r$) with connected deficiency graphs. Consequently, we prove that there exist pentagonal geometries PENT($k$,$r$) with deficiency graphs of girth at least 5 for $r \ge 13$, $r$ congruent to 1 modulo 4 when $k = 4$, and for $r \ge 200000$, $r$ congruent to 0 or 1 modulo 5 when $k = 5$. We conclude with a discussion of appropriately defined identifying codes for pentagonal geometries., Comment: 84 pages. Abstract changed. Lemmas and theorems in Sections 4 and 5 significantly improved. Many more directly constructed PENT(4,r). New section: Identifying codes for pentagonal geometries. The paper with the appendix omitted will be submitted to a journal

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. Here we generalize the concept by allowing the points not collinear with $x$ to form the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry., Comment: 143 pages, 1 figure. An abridged version, with the Appendix omitted, will be submitted to a journal