ABSTRACT In 1948 N. G. de Bruijn and P. Erdős [Proc. Akad. Wet. Amsterdam 51, 1277–1279 (1948; Zbl 0032.24405)] proved that every finite linear space on v points and with b lines fulfils the inequality b≥v, and the equality holds if the... more
ABSTRACT In 1948 N. G. de Bruijn and P. Erdős [Proc. Akad. Wet. Amsterdam 51, 1277–1279 (1948; Zbl 0032.24405)] proved that every finite linear space on v points and with b lines fulfils the inequality b≥v, and the equality holds if the linear space is a (possibly degenerate) projective plane. This result led to the problem of classifying finite linear spaces on v points and with b=v+s lines, s≥1. This paper contains the classification of finite linear spaces on v points and with b=v+4 lines.
Research Interests:
Research Interests:
Research Interests:
A set of type (m,n)K is a set of points of a finite linear space with property that each line of the linear space meets either m or n points of K. In this paper, sets of type (m,n) in a finite linear space with constant point degree q+1... more
A set of type (m,n)K is a set of points of a finite linear space with property that each line of the linear space meets either m or n points of K. In this paper, sets of type (m,n) in a finite linear space with constant point degree q+1 are defined. Then the author studies the (q+1)-regular finite spaces containing a set of type (m,n), and finite planar spaces whose planes pairwise intersect each other either in the empty-set or in a line and with a set of type (0,n). At the end of the article, the author obtains a new proof of a result of M. Tallini Scafati [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 53(1972), 71–81 (1973; Zbl 0267.50016)] on sets of type (0,n) in PG(r,q).
Research Interests:
In this note we investigate some properties of lines of maximal size in a finite linear space. Also, we give a new and unified proof of two theorems by Hanani (6), (and Varga (10)) and Melone (9).
Research Interests:
We show that (q+1)-arcs and classical ovoids of PG(3,q) are set-theoretic complete intersections.
In this note we give a short and geometric proof of a famous result on finite planar spaces which states that the number of planes is greater than or equal to the number of points, and that the equality holds if, and only if, the planar... more
In this note we give a short and geometric proof of a famous result on finite planar spaces which states that the number of planes is greater than or equal to the number of points, and that the equality holds if, and only if, the planar space is either a finite 3-dimensional projective space, or there are two disjoint lines containing all the points of the planar space, or all the points but one belong to a finite projective plane.
Research Interests:
Let (P,ℒ) be a finite linear space with v points and b lines. Moreover, let m and k denote the minimum point degree and the maximum line length of (P,ℒ), respectively. Extending a result of N. Durante [Discrete Math. 255, 71–80 (2002; Zbl... more
Let (P,ℒ) be a finite linear space with v points and b lines. Moreover, let m and k denote the minimum point degree and the maximum line length of (P,ℒ), respectively. Extending a result of N. Durante [Discrete Math. 255, 71–80 (2002; Zbl 1024.51008)], the author determines all linear spaces with b-v≤k. Furthermore, he shows that every linear space with b-v=1+v satisfies either b-v≤k or b-v≤m. Since recently he determined all linear spaces with b-v≤m, cf. [V. Napolitano, Discrete Math. 270, 207–224 (2003; Zbl 1032.51003)], he also gets a classification of all linear spaces with b-v=1+v.
Research Interests:
The author poves the following theorem for a finite planar space S. If S has no disjoint subplanes and if the number of planes on every line is constant, then the number of lines b is greater or equal to the number c of planes in S.... more
The author poves the following theorem for a finite planar space S. If S has no disjoint subplanes and if the number of planes on every line is constant, then the number of lines b is greater or equal to the number c of planes in S. Equality b=c holds if and only if S is either PG(4,q) or the complete graph K 5 .
Research Interests:
An 𝕃(n,d) is a linear space with constant point degree n+1, lines of size n and n-d, and with v=n 2 -d points. Denote by b=n 2 +n+z the number of lines of an 𝕃(n,d), then z≥0 and examples are known only if z=0,1 [K. Metsch, Linear spaces... more
An 𝕃(n,d) is a linear space with constant point degree n+1, lines of size n and n-d, and with v=n 2 -d points. Denote by b=n 2 +n+z the number of lines of an 𝕃(n,d), then z≥0 and examples are known only if z=0,1 [K. Metsch, Linear spaces with few lines (1991; Zbl 0744.51005)]. The linear spaces 𝕃(n,d) were introduced in [loc. cit.] in relation with some classification problems of finite linear spaces. In this note, starting from the symmetric configuration 45 7 of R. D. Baker [J. Comb. Theory, Ser. A 25, 193–195 (1978; Zbl 0433.05017)] we give an example of 𝕃(n,d), with n=7, d=4 and z=4.
Research Interests:
ABSTRACT A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q 2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is... more
ABSTRACT A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q 2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.
Research Interests:
The plane degree g_K(2) of a subset K of PG(3, q) is the greatest integer such that at least one plane intersecting K in exactly g_K(2) points exists. In this note, (q+1)-arcs of PG(3, q) (that is, twisted cubics when q is odd) are... more
The plane degree g_K(2) of a subset K of PG(3, q) is the greatest integer such that at least one plane intersecting K in exactly g_K(2) points exists. In this note, (q+1)-arcs of PG(3, q) (that is, twisted cubics when q is odd) are characterized as (q + 1)-sets of type (0, 1, s)1 of PG(3, q) of minimal plane degree.
Research Interests:
In this paper, sets of type (1,h) in a planar space whose planes pairwise intersect either in the empty-set or in a line, are studied where a set of type (m,n) is a set K of points of a planar space with the property that each plane of... more
In this paper, sets of type (1,h) in a planar space whose planes pairwise intersect either in the empty-set or in a line, are studied where a set of type (m,n) is a set K of points of a planar space with the property that each plane of the space meets K either in m or n points, and there are both planes intersecting K in m points and in n points.
Research Interests:
ABSTRACT In 1948 N. G. de Bruijn and P. Erdős [Proc. Akad. Wet. Amsterdam 51, 1277–1279 (1948; Zbl 0032.24405)] proved that every finite linear space on v points and with b lines fulfils the inequality b≥v, and the equality holds if the... more
ABSTRACT In 1948 N. G. de Bruijn and P. Erdős [Proc. Akad. Wet. Amsterdam 51, 1277–1279 (1948; Zbl 0032.24405)] proved that every finite linear space on v points and with b lines fulfils the inequality b≥v, and the equality holds if the linear space is a (possibly degenerate) projective plane. This result led to the problem of classifying finite linear spaces on v points and with b=v+s lines, s≥1. This paper contains the classification of finite linear spaces on v points and with b=v+4 lines.
Research Interests:
Research Interests:
ABSTRACT A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q 2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is... more
ABSTRACT A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q 2 + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.
Research Interests:
ABSTRACT In this article author study on the projective space PG(3,q) and he show that two nice results where K denote a set of type (q+1,n) 2 in PG(3,q) of size k: -If L and L ' two lines in K and π is a plane containing L and L... more
ABSTRACT In this article author study on the projective space PG(3,q) and he show that two nice results where K denote a set of type (q+1,n) 2 in PG(3,q) of size k: -If L and L ' two lines in K and π is a plane containing L and L ' then K=π· -If k=(q+1) 2 and K contains at least q-q pairwise skew lines then either K is a set of points of q+1 pairwise skew lines or q is square and K is the set of points of q-q pairwise skew lines.
Research Interests:
ABSTRACT Let m,nm,n be two positive integers, a subset KK of points of the finite desarguesian 3-dimensional projective space PG(3,q)PG(3,q) is of class [1,m,n]2[1,m,n]2 if every plane of PG(3,q)PG(3,q) intersects KK either in 1, or m or... more
ABSTRACT Let m,nm,n be two positive integers, a subset KK of points of the finite desarguesian 3-dimensional projective space PG(3,q)PG(3,q) is of class [1,m,n]2[1,m,n]2 if every plane of PG(3,q)PG(3,q) intersects KK either in 1, or m or n points. We shall present a result on (q2+q+1)(q2+q+1)-sets of points of PG(3,q)PG(3,q) of class [1,m,n]2[1,m,n]2 which implies a characterization of quadric cones of PG(3,q)PG(3,q), q odd.
Research Interests:
Research Interests:
Research Interests:
We introduce a new method to describe tactical (de-)compositions of symmetric configurations via block (0,1)-matrices with constant row and column sum having circulant blocks. This method allows us to prove the existence of an infinite... more
We introduce a new method to describe tactical (de-)compositions of symmetric configurations via block (0,1)-matrices with constant row and column sum having circulant blocks. This method allows us to prove the existence of an infinite class of symmetric configurations of type (2p2)p+s where p is any prime and s≤t is a positive integer such that t−1 is the greatest prime
Research Interests:
Research Interests:
In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (\kappa - 1) I_n + J_n - A A^{\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all... more
In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (\kappa - 1) I_n + J_n - A A^{\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $\kappa$, respectively.
Research Interests:
In this paper, a complete classification of subsets of points of PG(3, q) of type (3, q + 3) with respect to planes is given.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
ABSTRACT In this paper, a description of sets of points of PG(3, q) of type (q + 1, n) with respect to the planes is given.
Research Interests:
Let $k,l,m,n$, and $\mu$ be positive integers. A $\mathbb{Z}_\mu$--{\it scheme of valency} $(k,l)$ and {\it order} $(m,n)$ is a $m \times n$ array $(S_{ij})$ of subsets $S_{ij} \subseteq \mathbb{Z}_\mu$ such that for each row and column... more
Let $k,l,m,n$, and $\mu$ be positive integers. A $\mathbb{Z}_\mu$--{\it scheme of valency} $(k,l)$ and {\it order} $(m,n)$ is a $m \times n$ array $(S_{ij})$ of subsets $S_{ij} \subseteq \mathbb{Z}_\mu$ such that for each row and column one has $\sum_{j=1}^n |S_{ij}| = k $ and $\sum_{i=1}^m |S_{ij}| = l$, respectively. Any such scheme is an algebraic equivalent of a $(k,l)$-semi-regular bipartite voltage graph with $n$ and $m$ vertices in the bipartition sets and voltages coming from the cyclic group $\mathbb{Z}_\mu$. We are interested in the subclass of $\mathbb{Z}_\mu$--schemes that are characterized by the property $a - b + c - d\; \not \equiv \;0$ (mod $\mu$) for all $a \in S_{ij}$, $b \in S_{ih}$, $c \in S_{gh}$, and $d \in S_{gj}$ where $i,g \in {1,...,m}$ and $j,h \in {1,...,n}$ need not be distinct. These $\mathbb{Z}_\mu$--schemes can be used to represent adjacency matrices of regular graphs of girth $\ge 5$ and semi-regular bipartite graphs of girth $\ge 6$. For suitable $\...
We present three constructions which transform some symmetric config-uration K of type n k into new symmetric configurations of types (n + 1) k , or n k−1 , or ((λ − 1)µ) k−1 if n = λµ. Applying them to Desarguesian ellip-tic semiplanes,... more
We present three constructions which transform some symmetric config-uration K of type n k into new symmetric configurations of types (n + 1) k , or n k−1 , or ((λ − 1)µ) k−1 if n = λµ. Applying them to Desarguesian ellip-tic semiplanes, an infinite family of new configurations comes into being, whose types fill large gaps in the parameter spectrum of symmetric config-urations.
A (tactical) configuration of type (n r ,b k ) is a finite incidence structure consisting of a set of n points and a set of b lines such that (i) each line is incident with exactly k points and each point is incident with exactly r lines,... more
A (tactical) configuration of type (n r ,b k ) is a finite incidence structure consisting of a set of n points and a set of b lines such that (i) each line is incident with exactly k points and each point is incident with exactly r lines, (ii) two distinct points are incident with at most one line. If n=b (or equivalently r=k), the configuration is called symmetric and its type is indicated with the symbol n k . The deficiency of a symmetric configuration 𝒞 is d:=n-k 2 +k-1· The deficiency is zero if and only if 𝒞 is a finite projective plane. Well-known examples of symmetric configurations are projective planes. Indeed, any finite projective plane of order q is a symmetric configuration of type (q 2 +q+1) q+1 . Other examples include elliptic semiplanes. An elliptic semiplane of order ν is a configuration of type n ν+1 satisfying the following axioms of parallels: given a non-incident point-line pair (p,l), there exists at most one line l ' through p parallel to l (i.e., l and ...
ABSTRACT In this paper finite partially proper {0,1}-semiaffine planes of order n are studied and completely characterized. Finite partially {0}-semiaffine planes are completely classified and finite partially {1}-semiaffine planes are... more
ABSTRACT In this paper finite partially proper {0,1}-semiaffine planes of order n are studied and completely characterized. Finite partially {0}-semiaffine planes are completely classified and finite partially {1}-semiaffine planes are classified for b less than or equal to n^2 + n + 1.
Research Interests:
Research Interests:
Research Interests:
ABSTRACT Let PG (n,q) be the projective n-space over the Galois field GF(q). A k-cap in PG(n,q) is a set of k points such that no three of them are collinear. A k-cap is said to be complete if it is maximal with respect to set-theoretic... more
ABSTRACT Let PG (n,q) be the projective n-space over the Galois field GF(q). A k-cap in PG(n,q) is a set of k points such that no three of them are collinear. A k-cap is said to be complete if it is maximal with respect to set-theoretic inclusion. In this paper, using classical algebraic varieties, such as Segre varieties and Veronese varieties, some new infinite classes of caps are constructed.