Retrograde tracer injections in 29 of the 91 areas of the macaque cerebral cortex revealed 1,615 interareal pathways, a third of which have not previously been reported. A weight index (extrinsic fraction of labeled neurons [FLNe]) was determined for each area-to-area pathway. Newly found projections were weaker on average compared with the known projections; nevertheless, the 2 sets of pathways had extensively overlapping weight distributions. Repeat injections across individuals revealed modest FLNe variability given the range of FLNe values (standard deviation <1 log unit, range 5 log units). The connectivity profile for each area conformed to a lognormal distribution, where a majority of projections are moderate or weak in strength. In the G29 × 29 interareal subgraph, two-thirds of the connections that can exist do exist. Analysis of the smallest set of areas that collects links from all 91 nodes of the G29 × 91 subgraph (dominating set analysis) confirms the dense (66%) structure of the cortical matrix. The G29 × 29 subgraph suggests an unexpectedly high incidence of unidirectional links. The directed and weighted G29 × 91 connectivity matrix for the macaque will be valuable for comparison with connectivity analyses in other species, including humans. It will also inform future modeling studies that explore the regularities of cortical networks.

A Weighted and Directed Interareal Connectivity Matrix for Macaque Cerebral Cortex

Commutative rings I

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

The Mouse Cortical Connectome, Characterized by an Ultra-Dense Cortical Graph, Maintains Specificity by Distinct Connectivity Profiles

Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Maimani et al. [13] determined all isomorphism classes of finite commutative rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a commutative ring, we characterize all isomorphism classes of finite commutative rings whose total graph has genus two.

On the Genus of the Total Graph of a Commutative Ring

In this paper, we prove some general fixed point theorems, which generalize the result of Pant [4] for mappings in fuzzy metric spaces. Here we give the existence of a contractive definition which generates a fixed point, but does not force the mapping to be continuous at the fixed point.

Common fixed points of four mappings in a fuzzy metric space

The concept of dominating sets introduced by Ore and Berge, is currently receiving much attention in the literature of graph theory. Several types of domination parameters have been studied by imposing several conditions on dominating sets. Ore observed that the complement of every minimal dominating set of a graph with minimum degree at least one is also a dominating set. This implies that every graph with minimum degree at least one has two disjoint dominating sets. Recently several authors initiated the study of the cardinalities of pairs of disjoint dominating sets in graphs. The inverse domination number is the minimum cardinality of a dominating set whose complement contains a minimum dominating set. Motivated by the inverse domination number, there are studies which deals about two disjoint domination number of a graph.

Inverse Domination in Graphs

Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). denotes the complement of T Γ(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi [2]. In this article, we characterize all commutative rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for to be connected whenever is connected and prove that . It is also proved that if diam(T Γ(R)) = 2, then T Γ(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al. [1].

T. Tamizh Chelvam

Papers

On the Genus of the Total Graph of a Commutative Ring

Common fixed points of four mappings in a fuzzy metric space

Inverse Domination in Graphs

On the Total Graph and Its Complement of a Commutative Ring

Recent developments on the power graph of finite groups - a survey