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

Symmetric Colorings of the Hypercube and Hyperoctahedron

A self-complementary graph G is a subgraph of the complete graph K_n that is isomorphic to its complement. A self-complementary graph can be thought of as an edge 2-coloring of K_n that admits a color-switching automorphism. An automorphism of K_n that is color-switching for some edge 2-coloring is called a complementing automorphism. Complementing automorphisms for K_n have been characterized in the past by such authors as Sachs and Ringel. We are interested in extending this notion of self-complementary to other highly symmetric families of graphs; namely, the hypercube Q_n and its dual graph, the hyperoctahedron O_n. To that end, we develop a characterization of the automorphism group of these graphs and use it to prove necessary and sufficient conditions for an automorphism to be complementing. Finally, we use these theorems to construct a computer search algorithm which finds all self-complementary graphs in Q_n and O_n up to isomorphism for n=2,3,4

Symmetric Colorings of the Hypercube and Hyperoctahedron

A self-complementary graph G is a subgraph of the complete graph K_n that is isomorphic to its complement. A self-complementary graph can be thought of as an edge 2-coloring of K_n that admits a color-switching automorphism. An automorphism of K_n that is color-switching for some edge 2-coloring is called a complementing automorphism. Complementing automorphisms for K_n have been characterized in the past by such authors as Sachs and Ringel. We are interested in extending this notion of self-complementary to other highly symmetric families of graphs; namely, the hypercube Q_n and its dual graph, the hyperoctahedron O_n. To that end, we develop a characterization of the automorphism group of these graphs and use it to prove necessary and sufficient conditions for an automorphism to be complementing. Finally, we use these theorems to construct a computer search algorithm which finds all self-complementary graphs in Q_n and O_n up to isomorphism for n=2,3,4

Paley Graphs and Their Generalizations

To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We will study some important properties of the Paley graphs. In particular, we will show that the Paley graphs are connected, symmetric, and self-complementary. Also we will show that the Paley graph of order q is (q-1)/2 -regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors, which means that the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. In the first section of Chapter 3 we will see three examples of these generalizations and some of their basic properties. In the second section of Chapter 3 we will define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge if and only if there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m > 1, and we call them the m-Paley graphs. In the third section we will show that the m-Paley graph of order q is complete if and only if gcd(m, q - 1) = 1 and when d = gcd(m, q - 1) > 1, the m-Paley graph is q-1/d -regular. Also we will prove that the m-Paley graphs are symmetric but not self-complementary. We will show also that the m-Paley graphs of prime order are connected but the m-Paley graphs of order p^n, n > 1 are not necessary connected, for example they are disconnected if gcd(m, p^n - 1) =(p^n-1)/ 2.Comment: Master Thesi

Switching Equivalence in Symmetric n-Sigraphs-V

Introducing a new notion S-antipodal symmetric n-sigraph of a symmetric n-sigraph and its properties are obtained. Also giving the relation between antipodal symmetric n-sigraphs and S-antipodal symmetric n-sigraphs. Further, discussing structural characterization of S-antipodal symmetric n-sigraphs

Some identities for enumerators of circulant graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs of prime, twice prime and prime-squared orders. In particular, it is shown that the semi-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Keywords: circulant graph; cycle index; cyclic group; nearly doubled primes; Cunningham chain; self-complementary graph; tournament; mixed graphComment: 17 pages, 3 tables Categories: CO Combinatorics (NT Number Theory) Math Subject Class: 05C30; 05A19; 11A4