# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Counting symmetric configurations $v_3$. (English) Zbl 0940.05020
An incidence geometry $(P,{\cal B})$ consists of a set of $v$ points $P= \{p_1,\dots, p_v\}$ and a collection of $b$ blocks ${\cal B}= \{B_1,\dots, B_b\}$ such that $B_i\subseteq P$ for $i= 1,2,\dots, b$. The number of blocks containing a point $p\in P$ is called the degree, denoted by $[p]$. Blocks are also called lines and $|B|$ is called the length of the block $B$. A pair $(p,B)$ with $p\in B\in{\cal B}$ is called a flag. In this case, the point $p$ is said to lie on the line $B$. The line is said to pass through $p$. Additionally, in an incidence geometry any pair of points is joined by at most one line, i.e., $|B_i\cap B_j|\le 1$ for all $i, j\in\{1,\dots, b\}$, $i\ne j$. An incidence geometry $(p,{\cal B})$ is called configuration of type $(v_r,b_k)$ if (1) $|B_j|= k$ for $j= 1,2,\dots, b$ and (2) $[p_i]= r$ for $i= 1,2,\dots, v$. A configuration $(p,{\cal B})$ with $v= b$ is called symmetric. Its type is simply denoted by $v_r$ which is the same as $b_k$ because $k= r$ in this case. A configuration $C= (p,{\cal B})$ is called decomposable if it can be written as the union of two configurations $C_1$, $C_2$ on distinct point sets: $C_1= (p_1,{\cal B}_1)$, $C_2= (p_2,{\cal B}_2)$, and $P= P_1\cup P_2$ and ${\cal B}= {\cal B}_1\cup{\cal B}_2$. Indecomposable configurations are also called connected. The isomorphisms are structure preserving mappings. A triangle of a configuration consists of three points, say $a$, $b$, and $c$, such that the three pairs $\{a,b\}$, $\{b,c\}$, $\{c,a\}$ are contained in different blocks. A configuration that has no triangles is called a triangle-free configuration. An isomorphism $C\to C^d$ is called an anti-automorphism. A configuration which admits an anti-automorphism is called self-dual. An anti-automorphism of order $2$ is called a polarity. A configuration which admits a polarity is called self-polar. The authors give tables of configurations $v_3$ for $v\le 18$ and triangle-free configurations for $v\le 21$ together with some statistics about some properties of structures like transitivity, self-duality or self-polarity.

##### MSC:
 05B30 Other designs, configurations
Full Text:
##### References:
 [1] Betten, A.; Betten, D.: Regular linear spaces. Beiträge zur algebra und geometrie 38, No. 1, 111-124 (1997) · Zbl 0890.05014 [2] Brinkmann, G.: Fast generation of cubic graphs. J. graph theory 23, 139-149 (1996) · Zbl 0858.05093 [3] Coxeter, H. S. M.: Self-dual configurations and regular graphs. Bull. amer. Math. soc. 56, 413-455 (1950) · Zbl 0040.22803 [4] P. Dembowski, Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 44, Springer, New York, 1968. [5] H. Gropp, Configurations, in: C.J. Colburn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 253--255. [6] Gropp, H.: Configurations and graphs -- II. Discrete math. 164, 155-163 (1997) · Zbl 0868.05049 [7] Gropp, H.: Blocking set free configurations and their relations to digraphs and hypergraphs. Discrete math. 165/166, 359-370 (1997) · Zbl 0891.05016