Abstract Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of graph is fixed as complement some convex region in plane, and all other vertices are placed at mass center their neighbors, then this results unique embedding, planar. It also follows fairly quickly that minimizes sum squared edge lengths, conditional on face. However, it not clear ho...

‎the annihilator graph $ag(r)$ of a commutative ring $r$ is a simple undirected graph with the vertex set $z(r)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎in this paper we give the sufficient condition for a graph $ag(r)$ to be complete‎. ‎we characterize rings for which $ag(r)$ is a regular graph‎, ‎we show that $gamma (ag(r))in {1,2}$ and...

A graph is called equistable when there is a non-negative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We characterize those series-parallel graphs that are equistable, generalizing results of Mahadev, Peled and Sun about equistable outer-planar graphs.

The three well-known graph classes, planar graphs(P), series-parallel graphs (SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP ⊂ SP ⊂ P . It is known that box(G) ≤ 3 if G ∈ P and box(G) ≤ 2 if G ∈ OP . Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-par...

A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. A rooted plane graph has a designated outer vertex. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) denote the set of all triconnected rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we give an algorithm that enumerates all p...

A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles....

in this paper by using concepts of graph theory, forces method and substructure ing method for analysis of structures, some methods have been introduced for optimal analysis of structures in a large scale, using substructuring method. planar rigid frames assuming linear behavior and small deformations, have been optimal analyzed, by using forces method. thus, a fortran power station computation...