Karen Meagher

A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations @p,@s in S there is a point i@?{1,...,n} such that @p(i)=@s(i). Deza and Frankl [P. Frankl, M. Deza, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977) 352-360] proved that if S@?S(n) is intersecting then |S|@?(n-1)!. Further, Cameron and Ku [P.J. Cameron, C.Y. Ku, Intersecting families of permutations, European J. Combin. 24 (7) (2003) 881-890] showed that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result.

Covering arrays on graphs

Two vectors v, w in Zgn are qualitatively independent if for all pairs (a, b) ∈ Zg × Zg there is a position i in the vectors where (a, b) = (vi, wi). A covering array on a graph G, CA (n, G, g), is a |V(G)| × n array on Zg with the property that any two rows which correspond to adjacent vertices in G are qualitatively independent. The smallest possible n is denoted by CAN(G, g). These are an extension of covering arrays. It is known that CAN(Kω(G), g) ≤ CAN(G, g) ≤ CAN(Kχ(G), g). The question we ask is, are there graphs with CAN(G, g) < CAN(Kχ(G), g)? We find an infinite family of graphs that satisfy this inequality. Further we define a family of graphs QI(n, g) that have the property that there exists a CAN(n, G, g) if and only if there is a homomorphism to QI(n, g). Hence, the family of graphs QI(n, g) defines a generalized colouring. For QI(n, 2), we find a formula for both the chromatic and clique number and determine two necessary conditions for CAN (G, 2) < CAN(Kχ(G), 2). We also find the cores of all the QI(n, 2) and use this to prove that the rows of any covering array with g = 2 can be assumed to have the same number of 1s.

An Erdös--Ko--Rado Theorem for the Derangement Graph of

In this paper we prove an Erdos--Ko--Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group

{PGL}_3(q)

{PGL}_3(q)

Acting on the Projective Plane

, in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line. This gives the first evidence for the veracity of Conjecture 2 from K. Meagher and P. Spiga, An Erdos-Ko-Rado Theorem for the Derangement Graph of

A new proof for the Erdős-Ko-Rado theorem for the alternating group

{PGL}(2,q)

Minimum Number of Distinct Eigenvalues of Graphs

Acting on the Projective Line [J. Combin. Theory Ser. A, 118 (2011), pp. 532--544].

An Erdős-Ko-Rado theorem for finite 2-transitive groups

A subset S of the alternating group on n points is intersecting if for any pair of permutations π , ? in S , there is an element i ? { 1 , ? , n } such that π ( i ) = ? ( i ) . We prove if n ? 5 and S is intersecting, then | S | ? ( n - 1 ) ! 2 . Also, we prove that provided that n ? 5 , then the only sets S that meet this bound are the cosets of the stabilizer of a point of { 1 , ? , n } . These two results were first proven by Ku and Wong (2007), the proof given in this paper uses an algebraic method that is very different from the original proof.

On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

Compressed cliques graphs, clique coverings and positive zero forcing

We prove an analogue of the classical Erd?s-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set ? , we show that an intersecting set of maximal size in G has cardinality | G | / | ? | . This generalises and gives a unifying proof of some similar recent results in the literature.