A graph $G$ is called an $S$-graph if its periphery $Peri\left(G\right)$ is equal to its center eccentric vertices $Cep\left(G\right)$. Further, a graph $G$ is called a $D$-graph if $Peri\left(G\right)\cap Cep\left(G\right)=\varnothing$. We describe $S$-graphs and $D$-graphs for small radius. Then, for a given graph $H$ and natural numbers $r\ge 2$, $n\ge 2$, we construct an $S$-graph of radius $r$ having $n$ central vertices and containing $H$ as an induced subgraph. We prove an analogous existence theorem for $D$-graphs, too. At the end, we give some properties of $S$-graphs and $D$-graphs.

Let G = G1 ∪ G2 be the sum of two simple graphs G1,G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1,G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant det A(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope and effectiveness of our method we give some examples

This article presents two simple deterministic algorithms for finding the Minimum Spanning Tree in $O\left(\right|V|+|E\left|\right)$ time for any non-trivial class of graphs closed on graph minors. This applies in particular to planar graphs and graphs of bounded genus. Both algorithms run on a pointer machine and they require no a priori knowledge of the structure of the class except for its density. Edge weights are only compared.

The problem of embedding graphs into other graphs is much studied in the graph theory. In fact, much effort has been devoted to determining the conditions under which a graph G is a subgraph of a graph H, having a particular structure. An important class to study is the set of graphs which are embeddable into a hypercube. This importance results from the remarkable properties of the hypercube and its use in several domains, such as: the coding theory, transfer of information, multicriteria rule,...

The problem of embedding graphs into other graphs is much studied in the graph theory. In fact, much effort has been devoted to determining the conditions under which a graph G is a subgraph of a graph H, having a particular structure. An important class to study is the set of graphs which are embeddable into a hypercube. This importance results from the remarkable properties of the hypercube and its use in several domains, such as: the coding theory, transfer of information, multicriteria rule,...

In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of feasible merging components, which is related to classic LWF (Lauritzen, Wermuth and Frydenberg) Markov equivalence of chain graphs. These results are used to get a graphical characterisation of factorisation equivalence of classic chain graphs. As another example of the use...

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Gould and Hynds (1999) showed a well-known characterization of $G$ by its line graph $L\left(G\right)$ that has a 2-factor. In this paper, by defining two operations, we present a characterization for a graph $G$ to have a 2-factor in its line graph $L\left(G\right).$ A graph $G$ is called $N^2$-locally connected if for every vertex $x\in V\left(G\right),$$G[...$

A set of vertices of a graph G is a total dominating set if each vertex of G is adjacent to a vertex in the set. The total domination number of a graph Υt (G) is the minimum size of a total dominating set. We provide a short proof of the result that Υt (G) ≤ 2/3n for connected graphs with n ≥ 3 and a short characterization of the extremal graphs.

Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m\left(G\right)=ma{x}_{F\subseteq G}\left|E\left(F\right)\right|/\left|V\left(F\right)\right|$ and define the Ramsey density $m_{inf}(H,r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_{inf}(H,r)=(R-1)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál...

In this work, we prove that every closed, orientable 3-manifold M3 which is a two-fold covering of S3 branched over a link, has type six.