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In this paper we consider the possibility that groups S n w r S 2 , 4 ≤ n ≤ 63 , act flag-transitively on block designs with n 2 points. We rule out n ∈ { 7 , 11 , 15 , 19 , 23 , 31 , 35 , 43 , 47 , 59 } and confirm the required action for other n in the given range through performing the construction of base blocks of the desired designs. Our construction makes use of flag-transitive incidence structures and weakly flag-transitive incidence structures.

We consider problems involving rich homotheties in a set S of n points in the plane (that is, homotheties that map many points of S to other points of S ). By reducing these problems to incidence problems involving points and lines in R 3 , we are able to obtain refined and new bounds for the number of rich homotheties, and for the number of distinct equivalence classes, under homotheties, of k -element subsets of S , for any k ≥ 3 . We also discuss the extensions of these problems to three and higher dimensions.

A blowup of a graph Γ with respect to a graph Γ ′ is the graph Γ Γ ′ obtained from Γ by replacing every vertex u of Γ with a disjoint copy Γ u ′ of Γ ′ and attach every vertex of Γ u ′ to every vertex of Γ v ′ if there is an edge between u and v in Γ . Accordingly, a P -blowup of a graph Γ is a Γ ′ -blowup of Γ with Γ ′ being a P -graph for a graph-theoretical property P . We present combinatorial interpretations of Moore–Penrose inverses of incidence matrices of independent-blowups and clique-blowups of trees and apply them to obtain formulas for resistance distances between vertices of the corresponding blowup graphs as well as their Kirchhoff indices. Also, we give some results on arbitrary blowups of graphs as well.

A well-known technique to construct regular graphs with girth 5 is the amalgamation into the incidence graphs C q and L q , elliptic semiplanes of type C and L respectively, where q is a prime power. The case q odd has extensively been studied by means of amalgamations into L q . In this paper we provide new families of small regular graphs of girth 5 constructed by amalgamation into C q using finite fields of even order.

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques., Comment: Title is shortened. Some typos and small errors were corrected

A study of normal uniform multiplicative designs admitting groups of automorphisms which act semiregularly on points and blocks and with two orbits is made. We call these designs Bi-Cayley normal uniform multiplicative designs (BCNUMD). It is also determined when the incidence matrices of such designs have maximum determinant. An infinite family of BCNUMDs is constructed which gives rise to symmetric designs.

We study random constructions in incidence structures, and illustrate our techniques by means of a well-studied example from finite geometry. A maximal partial ovoid of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical lower bounds on the size of a maximal partial ovoid in a quadrangle of order ( s , t ) are linear in s . In this paper, in a wide class of quadrangles of order ( s , t ) we give a construction of a maximal partial ovoid of size at most s ? polylog ( s ) , which is within a polylogarithmic factor of theoretical lower bounds. The construction substantially improves previous quadratic upper bounds in quadrangles of order ( s , s 2 ) , in particular in the well-studied case of the elliptic quadrics Q - ( 5 , s ) .

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix N 2 (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular N 2 (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near ( v 2 ) is obtained for large orders v .

Let d , k , n be integers with 1 ≤ d k ≤ n − d . In Kantor (1972), Kantor proved that the incidence matrix of d -dimensional subspaces versus k -dimensional subspaces of an n -dimensional vector space has full row rank over the real number field R . In this paper, we generalize Kantor’s result to the attenuated space A ( n + l ; F q ) and show that the incidence matrix of d -dimensional subspaces versus k -dimensional subspaces of A ( n + l ; F q ) also has full row rank over R . As an application, we obtain upper bounds for the class dimension of association schemes based on attenuated spaces.

Linear codes arising from the row span over any prime field F"p of the incidence matrices of the odd graphs O"k for k>=2 are examined and all the main parameters obtained. A study of the hulls of these codes for p=2 yielded that for O"2 (the Petersen graph), the dual of the binary hull from an incidence matrix is the binary code from points and lines of the projective geometry PG"3(F"2), which leads to a correspondence between the edges and vertices of O"2 with the points and a collection of ten lines of PG"3(F"2), consistent with the codes. The study also gives the dimension, the minimum weight, and the nature of the minimum words, of the binary codes from adjacency matrices of the line graphs L(O"k).

A complete classification of 2-arc-transitive representations of primitive or bi-primitive groups of fourth-power-free order is given. The list consists of the following graphs: K"n, K"n","n-nK"2, K"r","r with r an odd prime, Peterson graph O"2, incidence and non-incidence graphs HD(11,5,2) and HD(11,6,2) of the Hadamard design on 11 points, some explicit coset graphs of two-dimensional projective groups PSL(2,p^l) with 1@?l@?2 and of Janko simple group J"1, and the standard double cover of these coset graphs. Moreover, a classification of primitive permutation groups of fourth-power-free order is given.

This paper deals with a particular class of coloured designs: the incidence vectors of all blocks of these designs have the same composition, and the same is true for the incidence vectors of all points. For this reason, we call these designs constant composition designs, or CC-designs for short. We will derive necessary and sufficient conditions on the existence and conclude the presentation with a collection of examples.

A $\{; ; 0, \pm 1\}; ; $-matrix $S$ is called a Siamese twin design sharing the entries of $I$, if $S=I+K-L$, where $I, K, L$ are non-zero $\{; ; 0, 1\}; ; $-matrices and both $I+K$ and $I+L$ are incidence matrices of symmetric designs with the same parameters. Let $p$ and $2p+3$ be prime powers and $p \equiv 3\ (mod\ 4)$. We construct a Siamese twin design with parameters $(4(p+1)^2, 2p^2 +3p+1, p^2 + p)$.

The incidence chromatic number of G, denoted by χi(G), is the least number of colors such that G has an incidence coloring. In this paper, we determine the incidence chromatic number of the powers of paths, trees, which are min{n,2k+1}, and Δ(T2)+1, respectively. For the square of a Halin graph, we give an upper bound of its incidence chromatic number.

This paper examines some of the properties of point-weight incidence structures, i.e. incidence structures for which every point is assigned a positive integer weight. In particular it examines point-weight designs with a design condition that stipulates that any two ''identical'' sets of t points must lie on the same number of blocks. We introduce a new class of designs with this property: row-sum designs, and examine the basic properties of row-sum point-weight designs and their similarities to classical (non-point-weight) designs and the point-weight designs of Horne [On point-weighted designs, Ph.D. Thesis, Royal Holloway, University of London, 1996].

Several problems are introduced. They are not “by the wayside” in the sense that they were once considered and later discarded. They are very much alive. They are by the wayside, because I believe they have not been worked on much, yet relate to the Classical Finite Geometries (that is, Lie incidence geometries of finite buildings) which I view as central.

The incidence structures known as (@a,@b)-geometries are a generalization of partial geometries and semipartial geometries. In a previous paper, a classification of (@a,@b)-geometries fully embedded in PG(n,q), q odd and @a>1, assuming that every plane of PG(n,q) containing an antiflag of S is either an @a-plane or a @b-plane, is given. The case that there is a so-called mixed plane and that @b=q+1, is also treated there. In this paper we will treat the case @b=q. This completes the classification of all proper (@a,@b)-geometries fully embedded in PG(n,q), q odd and @a>1, such that PG(n,q) contains at least one @a- or one @b-plane. For q even, some partial results are obtained.

Let X denote a finite set, k and n denote natural numbers and S 1 , …, S n denote subsets of X . Assume that no point of X lies in more than k of these subsets. In 1981 Beck and Fiala proved that there is a 2-coloring of X such that each of the subsets has discrepancy less than 2 k . This result has an interpretation as a theorem about incidence matrices and its generalization to real matrices (with essentially the same proof) is called the continuous Beck-Fiala theorem. We investigate the continuous version of the conjecture of Beck and Fiala that ‘less than 2 k ’ could be replaced by ‘less than or equal to k ’. For matrices with nonnegative entries, we show that the answer to the corresponding continuous problem is ‘no’, so the continuous Beck-Fiala theorem is optimal in this case. However our methods do not provide a counterexample to Beck and Fiala's original conjecture. On the other hand we show that the answer to the corresponding continuous problem is ‘yes’ when the dimension of the matrix that corresponds to n is 1, 2 or 3.

Given a complete graph K n on n nodes and a subset S of nodes, the cut δ( S ) defined by S is the set of edges of K n with exactly one endnode in S . A cut δ( S ) is an equicut if |S|=⌊ n 2 ⌋ or ⌈ n 2 ⌉ and an inequicut, otherwise. The cut cone C n (or inequicut cone IC n ) is the cone generated by the incidence vectors of all cuts (or inequicuts) of K n . The equicut polytope EP n , studied by Conforti et al. (1990), is the convex hull of the incidence vectors of all equicuts. We prove that IC n and EP n ‘inherit’ all facets of the cut cone C n , namely, that every facet of the cut cone C n yields (by zero-lifting) a facet of the inequicut cone IC m for n m 2 ⌋ and of EP m for m odd, m ⩾2 n +1. We construct several new classes of facets, not arising from C n , for the inequicut cone IC n and we describe its facial structure for n ⩽7.

Given a clutter L , we associate with it the covering polyhedron Q L , that is the dominant of the convex hull of incidence vectors of all the covers of L . In this paper we describe a binary composition operation, the anti-join, which combines a pair of clutters L 1 and L 2 to give a new clutter L . The anti-join operation is, in some sense, the “dual” of the join operation, introduced by Cunningham [10]. In fact, it has the property that the blocker of the clutter L obtained by joining two clutters L 1 and L 2, is the anti-join of the blockers of L 1 and L 2. For such an operation we show how the linear descriptions of the polyhedra Q( L 1) and Q( L 2) have to be combined to produce a linear description of the polyhedron Q L . Moreover, given a set F ⊆ {λ: 0⩽λ⩽1} such that {0, 1} ⊆ F and aϵ F if and only if (1−a)ϵ F , we define the F -property for covering polyhedra as a proper generalization of the Fulkerson property, to which it reduces for F ={0, 1} . We prove that the anti-join operation preserves the F -property. This implies the characterization of the coefficients of the facet-defining inequalities for the cycle and cocycle polyhedra associated with graphs noncontractible to the four-wheel W4.

In this paper a few relationships between a Decomposition Structure and its Incidence Coalgebra and Algebra are studied. In particular, some results about the so-called Isomorphism Problem for Incidence Algebras of Moebius Categories are generalized. Moreover, we determine conditions under which all endomorphisms and derivations of an Incidence Algebra are continuous with respect to the finite topology.

The nil radicals of an incidence algebra are determined. The prime ideal structure of the algebra and the unboundedness of the underlying partial order are related, leading to a notion of topological end for ordered sets.

A new method of constructing a series of partially balanced ternary (PBT) designs is presented. In the method, we have added the corresponding rows of incidence matrices of a BIB design and a PBIB design, both obtained from single initial block with at least one element in common between them. The BIB and PBIB designs above were obtained by method of differences. We have also constructed PBT designs and PB n-ary designs from a PBIB design alone based on NCm-scheme as well as from a group divisible PBIB design with smaller number of blocks and moderate block sizes.

ℓ-matrices are defined and shown to be the incidence matrices of certain matroids (lines). The well-known characterizations of binary matroids of W.T. Tutte and A. Lehman are proved along with a new characterization. The proofs use ℓ-matrices.

It is shown that the number of ‘true-isomorphism’ classes of expansion of a resolvable incidence structure to another incidence structure (‘at infinity’) can be expressed in terms of the cycle indices in the automorphism groups of the incidence structures. As an example all expansions the good divisible design (with three parallelisms) to another divisible design are determined.

In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U2,3(9) in PG(5, 2).

In this paper we determine, by two simple matrices, the expansion of the characteristic polynomial of the matrices A, A + Δ + , A + Δ − , Δ − − A, Δ + − A of a multidigraph, and we obtain some interesting consequences. We obtain new results and also new simple proofs of known theorems.

Asymptotics are obtained for the number of m×n non-negative integer matrices subject to the following constraints: (i) each row and each column sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled digraphs.

The cohomology groups Hn(M, C) are studied, where C is the incidence coalgebra of a locally finite partially ordered set P and where M is a C-C comodule depending on a convex coarsening of the given partial order on P. The case where P is a geometric lattice and the convex coarsening is just the equality relation is emphasized.

A configuration D with parameters (v, b, r, k) is an incidence structure (P, B, I), where P is a set of v ''points'', B is a set of b ''blocks'' and I is an ''incidence relation'' between points and blocks such that each point is incident with r blocks, and each blok is incident with k points. A block may be regarded as the subset of those points of p with which it is incident. A regular two component pairwise balanced design with parameters v, k"1,k"2r"1,r"2,1 is an incidence structure D = (P, B"1 @? B"2, I), where D"i = (P, B"i,I) is a configuration with parameters (v, b"i, r"i, k"i), i = 1,2, b"i = vr"i/k"i, and such that any two distinct points of P are contained in exactly one block of B"1 @? B"2. The configuration D"i is called the i^t^h component of D, i = 1,2. Hence by deleting the points of an oval S from a projective plane @? of even order q, we obtain a regular two component pairwise balanced design for which v=q^2-1, b"1=12q(q-1), b"2=12(q+1)(q+2), r"1=12q, r"2=12(q+2), k"1=q+1, k"2q-1. In this paper, we investigate the converse question and prove: Given a regular two component pairwise balanced design with the above parameters, then if q 6, the design can be embedded in a projective plane of order q.

The concept of a partial three-space is due to Laskar and Dunbar, and is a three-dimensional analogue of a partial geometry. Here we determine all partial three-spaces S for which the S-planes are planes of PG(n, q), for which the S-lines are all the lines contained in the S-planes, for which the S-points are all the points in the S-planes, and for which the incidence relation is that of PG(n, q). More generally, we determine all partial three-spaces S for which the S-lines are lines of PG(n,q), for which the S-points are all the points on these lines, and for which the incidence relation is that of PG(n, q).

There are four known finite projective planes of order 9. This paper reports the result of a computer search which shows that this list is complete. The computer search starts by generating all 283,657 non-isomorphic latin squares of order 8. Each latin square gives 27 columns of the incidence matrix. Another program attempts to complete each of these incidence matrices to 40 columns. Only 21 of them can be so completed, giving rise to 326 matrices of 40 columns. A third computer program attempts to complete the rest of the matrices. One of the 326 does not complete. The rest complete each to a unique matrix. An isomorphism testing program is then applied to the 325 complete matrices, creating a certificate for each matrix, as well as its collineation group. The certificates are then compared with the known planes and no new ones found. As a further evidence of the correctness of the computer programs, this paper also shows that the computer results are consistent with those expected by using information about the known planes and their associated latin squares.

It is known that the incidence algebra of a finite poset is not strongly simply connected if and only if its quiver contains a crown. We give a combinatorial condition on crowns which, if satisfied, forces the incidence algebra to be simply connected. The converse is not true, but we show that a simply connected incidence algebra which is not strongly simply connected always contains crowns satisfying this condition.

Let (P,⩽) be a locally finite partially ordered set. Let K(x,y;z) be a function of x,y,z∈P . We define the K -convolution of incidence functions f and g as (f★g)(x,y)= ∑ x⩽z⩽y f(x,z)g(z,y)K(x,y;z). We define two transformations on the set of incidence functions, which serve as logarithm and exponential operators under the K -convolution, give their basic properties and apply them in finding solutions for the functional equations f (r) =g , f (r) =fg and f★g=h in f , where f (r) denotes the r th iterate of f with respect to the K -convolution. These results cast some known results on arithmetical functions in the poset-theoretic framework.

The theory of reduced incidence algebras of binomial posets furnishes a unified treatment of several types of generating functions that arise in enumerative combinatorics. Using this theory as a tool, we study ‘reduced covering algebras’ of binomial lattices and show that they are isomorphic to various algebras of q-binomial generating functions for certain modular binomial lattices.

A new decomposition of strongly unimodular is given.