Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

Assessment of the Level of Integration and State of the Organization’s IT Infrastructure

The paper considers a method for assessing the state of an organization’s IT infrastructure using graph theory. The description of the main parameters which can be used to describe the state and assessment of infrastructure is given, the ways of their application are determined.

Altered Functional Network in Infants With Profound Bilateral Congenital Sensorineural Hearing Loss: A Graph Theory Analysis

Functional magnetic resonance imaging (fMRI) studies have suggested that there is a functional reorganization of brain areas in patients with sensorineural hearing loss (SNHL). Recently, graph theory analysis has brought a new understanding of the functional connectome and topological features in central neural system diseases. However, little is known about the functional network topology changes in SNHL patients, especially in infants. In this study, 34 infants with profound bilateral congenital SNHL and 28 infants with normal hearing aged 11–36 months were recruited. No difference was found in small-world parameters and network efficiency parameters. Differences in global and nodal topologic organization, hub distribution, and whole-brain functional connectivity were explored using graph theory analysis. Both normal-hearing infants and SNHL infants exhibited small-world topology. Furthermore, the SNHL group showed a decreased nodal degree in the bilateral thalamus. Six hubs in the SNHL group and seven hubs in the normal-hearing group were identified. The left middle temporal gyrus was a hub only in the SNHL group, while the right parahippocampal gyrus and bilateral temporal pole were hubs only in the normal-hearing group. Functional connectivity between auditory regions and motor regions, between auditory regions and default-mode-network (DMN) regions, and within DMN regions was found to be decreased in the SNHL group. These results indicate a functional reorganization of brain functional networks as a result of hearing loss. This study provides evidence that functional reorganization occurs in the early stage of life in infants with profound bilateral congenital SNHL from the perspective of complex networks.

Subgraph densities in a surface

Given a fixed graph H that embeds in a surface $\Sigma$ , what is the maximum number of copies of H in an n-vertex graph G that embeds in $\Sigma$ ? We show that the answer is $\Theta(n^{f(H)})$ , where f(H) is a graph invariant called the ‘flap-number’ of H, which is independent of $\Sigma$ . This simultaneously answers two open problems posed by Eppstein ((1993) J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, if H is a $K_{3,t}$ minor-free graph, then the maximum number of copies of H in an n-vertex $K_{3,t}$ minor-free graph G is $\Theta(n^{f'(H)})$ , where f′(H) is a graph invariant closely related to the flap-number of H. Finally, when H is a complete graph we give more precise answers.