The generalized pentagonal geometry PENT($k$,$r$,$w$) is a partial linear space where all lines are attached to the points in $k$ and all points are attached to $r The set of points attached to the line of $ and, for each point, $x$, not collinear with $x$, is the point set of the Steiner system $S(2,k,w)$ whose blocks are the lines of the geometry form the If $w = k$, the structure is called a pentagon geometry and is denoted by PENT($k$,$r$). The starvation graph of PENT($k$,$r$,$w$) has the points of the geometry as its vertices, and if $x$ and $y$, then exactly between $x$ and $y$ It has an edge. $ is not collinear.

Our main aim is to explore the generalized pentagon geometry PENT($k$,$r$,$w$). Here the deficiency graph has a girth of 4. A number of theorems about existence spectra for various values ​​of $k = 3$ and $w$. In addition, we present several new PENT(4,$r$) (including PENT(4,25)) and PENT(5,$r$) with connected deficiency graphs. Thus, for $k = 4$, the pentagon geometry PENT($k$, $r$) exists. For $r \ge 200000$ $r$ matches 0 or 1 modulo 5 if $k = 5$. Finally, we discuss a well-defined identification code for the pentagon geometry.

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