An interval graph is not a comparability graph

The asymptotic probability that a random graph is a unit interval graph, indifference graph, or proper interval graph

groups for which the noncommuting graph is a split graph

the noncommuting graph $nabla (g)$ of a group $g$ is asimple graph whose vertex set is the set of noncentral elements of$g$ and the edges of which are the ones connecting twononcommuting elements. we determine here, up to isomorphism, thestructure of any finite nonabeilan group $g$ whose noncommutinggraph is a split graph, that is, a graph whose vertex set can bepartitioned into two sets such t...

The world is not an asymmetric graph

1. Might the world be a mathematical structure? My question is not: might the world have a mathematical structure, but might it be identical with one? There are two general ways of interpreting this question. If it is whether, given that mathematical structures are abstract objects, the world is an abstract object, the answer has to be negative. Think of the world as wholly concrete (space, tim...

The Dimension of a Comparability Graph

Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph. In this paper we answer Bogart's question in the affirmative. The proof involves a characterization of the class of comparability graphs defined by Aigner and Prin...

Frucht Graph is not Hyperenergetic

If 1 2 , ,..., p    are the eigen values of a p-vertex graph , the energy of is G G 1 ( ) p i i E G     . If , then is said to be hyperenergetic. We show that the Frucht graph, the graph used in the proof of well known Frucht’s theorem, is not hyperenergetic. Thus showing that every abstract group is isomorphic to the automorphism group of some non-hyperenergetic graph. AMS Mathematics S...