A graph X is said to be distance–balanced if for any edge uv of X , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance–balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k + 1 from u a...

background: due to the wholesome benefits of nuts increased consumption of them has been recommended. however, because of nut’s high energy density, the role of them in the treatment of overweight and obesity is vague. this current clinical trial study aims to investigate the effects of a balanced hypocaloric almond-enriched diet (aed) (almond group) in comparison to a balanced hypocaloric nu...

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with u vertices and e edges if and only if I s u 1 s e s (2"). This result is applied to a classic question of Erdos and Renyi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem is also solved.

A biased graph consists of a graph T and a subclass B of the polygons of T, such that no theta subgraph of T contains exactly two members of B. A subgraph is balanced when all its polygons belong to B. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph ha...

In this paper, we first collect the earlier results about some graph operations and then we present applications of these results in working with chemical graphs.

A graph G is locally irregular if adjacent vertices of G have distinct degrees. An edge colouring of G is locally irregular if each of its colours induces a locally irregular subgraph of G. The irregular chromatic index of G refers to the least number of colours used by a locally irregular edge colouring of G (if any). We propose a linear-time algorithm for determining the irregular chromatic i...

I have lived in Michigan nearly all my life, having been brought up in Sault Ste. Marie, educated at Michigan State University, studying at the University of Michigan, and employed by Western Michigan University . I've enjoyed graph theory from the day I heard my eventual thesis advisor E. A . Nordhaus speak on it. I've had eleven doctoral students, the most recent being Ortrud Oellermann, and ...

A (d − 1)-dimensional simplicial complex is called balanced if its underlying graph admits a proper d-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for normal pseudomanifolds and characterize the case of equality; we introduce and ch...

For a graph G = (V,E) of even order, a partition (V1, V2) of the vertices is said to be perfectly balanced if |V1| = |V2| and the numbers of edges in the subgraphs induced by V1 and V2 are equal. For a base graph H define a random graph G(H, p) by turning every non-edge of H into an edge and every edge of H into a nonedge independently with probability p. We show that for any constant ǫ there i...