EXTENDING POTOČNIK AND ŠAJNA'S CONDITIONS ON VERTEX-TRANSITIVE SELF-COMPEMENTARY k-HYPERGRAPHS

Abstract : Let l be a positive integer, k = 2l or k = 2l + 1 and let n be a positive integer with $n \equiv 1$ (mod $2^{l+1}$). Potocnik and Sajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have $p^{n_{(p)}} \equiv 1 \pmod {2^{l+1}}$ (where $n_{(p)}$ denotes the largest integer $i$ for which $p^i$ divides $n$). Here we extend their result to any integer k and a larger class of integers n

Constructing Regular Self-complementary Uniform Hypergraphs

AMS Subject Classication Codes: 05C65, 05B05 05E20, 05C85.In this paper, we examine the possible orders of t-subset-regular self-complementary k-uniform hypergraphs, which form examples of large sets of two isomorphic t-designs. We reformulate Khosrovshahi and Tayfeh-Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1-subset-regular self-complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1.https://onlinelibrary.wiley.com/doi/abs/10.1002/jcd.2028

Brick assignments and homogeneously almost self-complementary graphs

AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K2n−nK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices

Self-Complementary Hypergraphs

In this thesis, we survey the current research into self-complementary hypergraphs, and present several new results. We characterize the cycle type of the permutations on n elements with order equal to a power of 2 which are k-complementing. The k-complementing permutations map the edges of a k-uniform hypergraph to the edges of its complement. This yields a test to determine whether a finite permutation is a k-complementing permutation, and an algorithm for generating all self-complementary k-uniform hypergraphs of order n, up to isomorphism, for feasible n. We also obtain an alternative description of the known necessary and sufficient conditions on the order of a self-complementary k-uniform hypergraph in terms of the binary representation of k. We examine the orders of t-subset-regular self-complementary uniform hyper- graphs. These form examples of large sets of two isomorphic t-designs. We restate the known necessary conditions on the order of these structures in terms of the binary representation of the rank k, and we construct 1-subset-regular self-complementary uniform hypergraphs to prove that these necessary conditions are sufficient for all ranks k in the case where t = 1. We construct vertex transitive self-complementary k-hypergraphs of order n for all integers n which satisfy the known necessary conditions due to Potocnik and Sajna, and consequently prove that these necessary conditions are also sufficient. We also generalize Potocnik and Sajna's necessary conditions on the order of a vertex transitive self-complementary uniform hypergraph for certain ranks k to give neces- sary conditions on the order of these structures when they are t-fold-transitive. In addition, we use Burnside's characterization of transitive groups of prime degree to determine the group of automorphisms and antimorphisms of certain vertex transitive self-complementary k-uniform hypergraphs of prime order, and we present an algorithm to generate all such hypergraphs. Finally, we examine the orders of self-complementary non-uniform hypergraphs, including the cases where these structures are t-subset-regular or t-fold-transitive. We find necessary conditions on the order of these structures, and we present constructions to show that in certain cases these necessary conditions are sufficient.University of OttawaDoctor of Philosophy in Mathematic

Observations on graph invariants with the Lovász ϑ-function

This paper delves into three research directions, leveraging the Lovász $\vartheta$-function of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that determine the capacity for two infinite subclasses of strongly regular graphs, and extending prior results. The second part explores cospectral and nonisomorphic graphs, drawing on a work by Berman and Hamud (2024), and it derives related properties of two types of joins of graphs. For every even integer such that $n \geq 14$, it is constructively proven that there exist connected, irregular, cospectral, and nonisomorphic graphs on $n$ vertices, being jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, while also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lovász $\vartheta$-functions. The third part focuses on establishing bounds on graph invariants, particularly emphasizing strongly regular graphs and triangle-free graphs, and compares the tightness of these bounds to existing ones. The paper derives spectral upper and lower bounds on the vector and strict vector chromatic numbers of regular graphs, providing sufficient conditions for the attainability of these bounds. Exact closed-form expressions for the vector and strict vector chromatic numbers are derived for all strongly regular graphs and for all graphs that are vertex- and edge-transitive, demonstrating that these two types of chromatic numbers coincide for every such graph. This work resolves a query regarding the variant of the $\vartheta$-function by Schrijver and the identical function by McEliece et al. (1978). It shows, by a counterexample, that the $\vartheta$-function variant by Schrijver does not possess the property of the Lovász $\vartheta$-function of forming an upper bound on the Shannon capacity of a graph. This research paper also serves as a tutorial of mutual interest in zero-error information theory and algebraic graph theory