Graphs Generated by Measures

Abstract

In this paper, a graph is assigned to any probability measure on the -algebra of Borel sets of a topological space. Using this construction, it is proved that given any number (finite or infinite) there exists a nonregular graph such that its clique, chromatic, and dominating number equals .

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

“…Researchers try to define a graph illustration for some kind of mathematical aspects. For example see [3] in the lattice literature, [13] in the measure literature and [16] in the real-valued continuous functions literature. The study of translating graph properties to algebraic properties is an interesting subject for mathematicians.…”

On Beck's Coloring for Measurable Functions

“…Beck conjectured that χ(G) = ω(G) for the zero divisor graph of R and an arbitrary ring R. But Anderson and Naseer [5] have shown that this is not the case in general, namely they presented an example of a commutative local ring R with 32 elements for which χ(G) > ω(G). After that lots of authors have worked on such graphs and found classes of graphs on which Beck's conjecture holds (see for instance [1,2,3,6,7,8,9,14,15,17]).…”

A little more on the zero-divisor graph and the annihilating-ideal graph of a reduced ring

“…The researchers tried to define a graph illustration for some kind of mathematical aspects. For example [3] in the lattice literature, [12] in the measure literature, [16] in topology literature and [13] in the linear algebra. The study of translating graph properties to algebraic properties is an interesting subject for mathematicians.…”