Let G be a simple balanced bipartite graph on 2n vertices, δ = δ(G)/n, and ρ0 = δ+ √ 2δ−1 2 . If δ ≥ 1/2 then G has a ⌊ρ0n⌋-regular spanning subgraph. The statement is nearly tight.

The dilation of a geometric graph is the maximum, over all pairs of points in the graph, of the ratio of the Euclidean length of the shortest path between them in the graph and their Euclidean distance. We consider a generalized version of this notion, where the nodes of the graph are not points but axis-parallel rectangles in the plane. The arcs in the graph are horizontal or vertical segments...

We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady state...

A graph G = (V,E) is called a probe graph of graph class G if V can be partitioned into two sets P (probes) and N (nonprobes), where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain nonprobes. A graph is distance hereditary if the distance between any two vertices remains the same in every connected induced subgraph. Distancehereditary graph...

let g be a connected simple (molecular) graph. the distance d(u, v) between two vertices u and v of g is equal to the length of a shortest path that connects u and v. in this paper we compute some distance based topological indices of h-phenylenic nanotorus. at first we obtain an exactformula for the wiener index. as application we calculate the schultz index and modified schultz index of this ...

Nash-Williams’ well-balanced orientation theorem [11] is extended for orienting several graphs simultaneously. We prove that if G1, ..., Gk are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation ~ G such that the inherited orientations ~ Gi of Gi are well-balanced for all 1 ≤ i ≤ k. We also have new results about simultaneous well-balanced orientations of non-...

We will be considering sets of n-tuples over an alphabet A, in two important cases: A ¡ ¢ 0£ 1¤ (binary code); n¤ , all entries of each word distinct (set of permutations). We often impose closure conditions on these sets, as follows: A binary code is linear if it is closed under coordinatewise addition mod 2. A set of permutations is a group if it is closed under composition. x£ yïs the number...

An algebraic distance graph is defined to be a graph with vertices in En in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that ( 1 ) if all the sides of a convex polygon T which is inscribed in a c...

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph Kn. Then, traverse the ordered edges and add each traversed edge to an (initially empty) evolving graph unless its addition creates a triangle. We study the evolving graph at around the time where Θ(n) edges have been traversed for any fixed ε ∈ (0, ...

A graph G is said to be pancyclic if it contains cycles of all lengths from 4 to |V (G)| in G. For any two vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path joining u and v lies on the cycle. Let G be a bipartite graph. For any two vertices u and v in G, a cycle C is called a balanced cycle between u and v if dC(u, v) = max{dC(x, y) | dG(x, u) and dG(y, v...