Self-complementary graphs and Ramsey numbers Part I: the decomposition and construction of self-complementary graphs

AbstractA new method of studying self-complementary graphs, called the decomposition method, is proposed in this paper. Let G be a simple graph. The complement of G, denoted by Ḡ, is the graph in which V(Ḡ)=V(G); and for each pair of vertices u,v in Ḡ,uv∈E(Ḡ) if and only if uv∉E(G). G is called a self-complementary graph if G and Ḡ are isomorphic. Let G be a self-complementary graph with the vertex set V(G)={v1,v2,…,v4n}, where dG(v1)⩽dG(v2)⩽⋯⩽dG(v4n). Let H=G[v1,v2,…,v2n],H′=G[v2n+1,v2n+2,…,v4n] and H∗=G−E(H)−E(H′). Then G=H+H′+H∗ is called the decomposition of the self-complementary graph G.In part I of this paper, the fundamental properties of the three subgraphs H,H′ and H∗ of the self-complementary graph G are considered in detail at first. Then the method and steps of constructing self-complementary graphs are given. In part II these results will be used to study certain Ramsey number problems (see (II))

Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA

Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints

Rough Neutrosophic Digraphs with Application

A rough neutrosophic set model is a hybrid model which deals with vagueness by using the lower and upper approximation spaces. In this research paper, we apply the concept of rough neutrosophic sets to graphs. We introduce rough neutrosophic digraphs and describe methods of their construction. Moreover, we present the concept of self complementary rough neutrosophic digraphs. Finally, we consider an application of rough neutrosophic digraphs in decision-making

Dually vertex-oblique graphs

AbstractA vertex with neighbours of degrees d1⩾⋯⩾dr has vertex type (d1,…,dr). A graph is vertex-oblique if each vertex has a distinct vertex type (no graph can have distinct degrees). Schreyer et al. [Vertex-oblique graphs, same proceedings] have constructed infinite classes of super vertex-oblique graphs, where the degree types of G are distinct even from the degree types of G¯.G is vertex-oblique iff G¯ is; but G and G¯ cannot be isomorphic, since self-complementary graphs always have non-trivial automorphisms. However, we show by construction that there are dually vertex-oblique graphs of order n, where the vertex-type sequence of G is the same as that of G¯; they exist iff n≡0 or 1(mod4),n⩾8, and for n⩾12 we can require them to be split graphs.We also show that a dually vertex-oblique graph and its complement are never the unique pair of graphs that have a particular vertex-type sequence; but there are infinitely many super vertex-oblique graphs whose vertex-type sequence is unique

The Lie h-Invariant Conformal Field Theories and the Lie h-Invariant Graphs

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characterization of these graphs is another aspect of the recently observed Lie group-theoretic structure of graph theory.Comment: 38p