Finite Projective Plane Of Order Research Articles

Even Maps, the Colin de Verdière Number and Representations of Graphs

Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on σ(G) which is, in general, tight.Equality between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.

Generalizing Korchmáros—Mazzocca Arcs

In this paper, we generalize the so-called Korchmáros—Mazzocca arcs, that is, point sets of size q + t intersecting each line in 0, 2 or t points in a finite projective plane of order q. For t ≠ 2, this means that each point of the point set is incident with exactly one line meeting the point set in t points.In PG(2, pn), we change 2 in the definition above to any integer m and describe all examples when m or t is not divisible by p. We also study mod p variants of these objects, give examples and under some conditions we prove the existence of a nucleus.

Laplacian Integral of Particular Steiner System

The notion of a hypergraph is motivated by a graph. In graph, every edge contains of two vertices. However, a hypergraph edges contains more than two vertices. In this article use hyperedge to mention edge of hypergraph. A finite projective plane of order n, denoted by , is a linear intersecting hypergraph. In this research finite projective plane order is Laplacian integral.

Small stopping sets in finite projective planes of order q

A configuration C in a (finite) incidence structure is a subset C of blocks. If every point on a block of C belongs to at least one other block of C, then C is called stopping set (or equivalently full configuration). If smin(q) is the minimal size of a stopping set in a finite projective plane of odd order q, then either smin(q) ≥ q + 5 if $3\not\vert q$ or smin(q) ≥ q + 3 if 3|q. In this note, we prove that smin(q) ≥ q + 5 for any odd q ≠ 3. If q = 3, then smin(3) = 6 and a stopping set of minimal size 6 in PG(2, 3) is the dual set of the symmetric difference of two lines. Also, we study stopping sets of size q + 4 in a finite projective plane of order q.

An elementary, geometric proof of the non-existence of a projective plane of order 6.

We present a fairly elementary, self-contained proof of the nonexistence of a finite projective plane of order $6$. Our approach is motivated by theory of binary codes but does not appeal to it directly.

Upper Bounds for k-Tuple (Total) Domination Numbers of Regular Graphs

Let k be a positive integer. A k-tuple (total) dominating set in a graph G is a subset S of its vertices such that any vertex v in G has at least k vertices inside S among its neighbors and v itself (resp. among its neighbors only). The minimum size of a k-tuple (total) dominating set for G is called the k-tuple (total) domination number of G. It is shown in this article that the k-tuple total domination number of a connected $$(k+1)$$-regular graph with n vertices is at most $$\frac{k^2+k-1}{k^2+k}n$$, unless it is the incident graph of a finite projective plane of order k, where this number is $$\frac{k^2+k}{k^2+k+1}n$$. We also show that the k-tuple domination number of a connected k-regular graph with n vertices is at most $$\frac{k^2-1}{k^2}n$$, unless it is a Moore graph of diameter 2, where this number is $$\frac{k^2}{k^2+1}n$$.

On the Embedding of an Arc into Cubic Curves in a Finite Projective Plane of Order Seven

The main aims of this research are to find the stabilizer groups of cubic curves over a finite field of order 7 and studying the properties of their groups and then constructing the arcs of degree 2 which are embedding in cubic curves of even size which are considering as the arcs of degree 3. Also drawing all these arcs.

Largest regular multigraphs with three distinct eigenvalues

We deal with connected k-regular multigraphs of order n that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given k. For k=2,3,7, the Moore graphs are largest. For k≠2,3,7,57, we show an upper bound n≤k2−k+1, with equality if and only if there exists a finite projective plane of order k−1 that admits a polarity.

Cubic Arcs in the Projective Plane Over a Finite Field of Order Twenty Three

In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in, studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.

Minimal Multiple Blocking Sets

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

Classification of Arcs in Finite Projective Plane of Order Sixteen

The aim of the paper is to classify certain geometric structures, called arcs. The main computing tool is the mathematical programming language GAP. In the plane PG(2,16),the important arcs are called complete and are those that cannot be increased to a larger arc. So far, all arcs up to size eighteen have been classified. Each of these arcs gives rise to an error-correcting code that corrects the maximum possible number of errors for its length.

The Finite Projective Plane and the 5-Uniform Linear Intersecting Hypergraphs with Domination Number Four

The matching number \(\alpha '(H)\) of a hypergraph H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A dominating set in H is a subset D of vertices of H such that for every \(v\in V(H)\setminus D\) there exists \(u\in D\) such that u and v lie in an edge of H, and the domination number of H, denoted by \(\gamma (H)\), is the minimum cardinality of a dominating set in H. It was shown that a Ryser-like inequality \(\gamma (H)\le (r-1)\alpha '(H)\) holds for hypergraphs H of rank r. In particular, for intersecting hypergraphs H of rank r, \(\gamma (H)\le r-1\), since \(\alpha '(H)=1\). The linear intersecting hypergraphs of rank \(2\le r\le 4\) achieving the equality \(\gamma (H)=r-1\) have been characterized. In this paper we show that all the 5-uniform linear intersecting hypergraphs H with equality \(\gamma (H)=r-1\) are generated by the finite projective plane of order three.

On the Embedding of an Arc Into a Cubic Curves in a Finite Projective Plane of Order Five

The main aims of this research is to find the stabilizer groups of a cubic curves over a finite field of order , studying the properties of their groups and then constructing the arcs of degree which are embedding in a cubic curves of even size which are considering as the arcs of degree . Also drawing all these arcs.

Character tables and the problem of existence of finite projective planes

AbstractWe use the connection between positive definite functions and the character table of the symmetric group S6 to give a short new proof of the nonexistence of a finite projective plane of order 6. For higher orders, like 10 and 12, the method seems to be inconclusive as of now, but could be a basis of further research.

Finite Projective Planes and the Delsarte LP-Bound

We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.

Saturating sets in projective planes and hypergraph covers

Let Πq be an arbitrary finite projective plane of order q. A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈3qlnq⌉+⌈(q+1)∕2⌉. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.

Segment Combining and Patching for Short Packet Forward Error Control Coding

Modern networks are popular with a large number of short packets generated by many light weight, low power devices. We propose a new approach, segment combining and patching (SCP), for forward error control coding suitable for short packets, where the encoder logically divides a full packet into a number of partially overlapped subpackets based on a Finite Projective Plane of order m, and transmits the full packet n times. Upon receiving all n replicas of a full packet, the decoder generates an additional combined packet using Maximum Ratio Combining. By selecting non-faulty subpackets through Cyclic Redundancy Checks, both inter-cluster and intra-cluster patching are then performed on the n + 1 clusters of subpackets until no new non-faulty subpackets are generated. Simulation results in both AWGN and block fading channels demonstrate that the SCP is an effective approach for forward error control coding for short packet communication in embedded networks, sensor networks, and so on. Lastly, we provide a theory of SCP to show how it works.

Counting arcs in projective planes via Glynn’s algorithm

An n-arc in a projective plane is a collection of n distinct points in the plane, no three of which lie on a line. Formulas counting the number of n-arcs in any finite projective plane of order q are known for $$n \le 8$$ . In 1995, Iampolskaia, Skorobogatov, and Sorokin counted 9-arcs in the projective plane over a finite field of order q and showed that this count is a quasipolynomial function of q. We present a formula for the number of 9-arcs in any projective plane of order q, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin’s formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.

Some Properties from Construction of Finite Projective Planes of Order 2 and 3

In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system S(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.

Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.