An acyclic orientation of a graph is an assignment of a direction to each edge in a way that does not form any directed cycles. Acyclic orientations of a complete bipartite graph are in bijection with a class of matrices called lonesum matrices, which can be uniquely reconstructed from their row and column sums. We utilize this connection and other properties of lonesum matrices to determine an analytic form of the generating function for the length of the longest path in an acyclic orientation on a complete bipartite graph, and then study the distribution of the length of the longest path when the acyclic orientation is random. We use methods of analytic combinatorics, including analytic combinatorics in several variables (ACSV), to determine asymptotics for lonesum matrices and other related classes. / Includes bibliography. / Dissertation (PhD)--Florida Atlantic University, 2021. / FAU Electronic Theses and Dissertations Collection

Lattices of hereditary properties of nite graphs have been extensively studied. We investigate the lattice L of induced-hereditary properties of countable graphs. Of interest to us will be some of the members of L. Much of our focus will be on hom-properties. We analyze their behaviour and consider their link to solving the long standing Hedetniemi Conjecture. We then discuss universal graphs and construct a universal graph for hom-properties. We then use these universal graphs to prove a theorem by Szekeres and Wilf. Lastly we off er a new proof of a theorem by Du ffus, Sands and Woodrow. / Dissertation (MSc)--University of Pretoria, 2013. / Mathematics and Applied Mathematics / Unrestricted

In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian.

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No description available.

The Ramsey number $R(r, b)$ is the least positive integer such that every edge 2-coloring of the complete graph $K_{R(r, b)}$ with colors red and blue either embeds a red $K_r$ or a blue $K_b$. We explore various methods to find lower bounds on $R(r,b)$, finding new results on fibrations and semicirculant graphs. Then, generalizing the Ramsey number to graphs other than complete graphs, we flesh out the missing details in the literature on a theorem that completely determines the generalized Ramsey number for cycles.

Fine-grained descriptions of brain connectivity are required to understand how neural information is processed and relayed across spatial scales. Prior investigations of the mouse brain connectome have employed discrete anatomical parcellations, limiting spatial resolution and potentially concealing network attributes critical to connectome organization. In this work, we provide a voxel-level description of the network and hierarchical structure of the directed mouse connectome, unconstrained by regional partitioning. We found that hub regions and core network components of the voxel-wise mouse connectome exhibit a rich topography encompassing key cortical and subcortical relay regions. We also typified regional substrates based on their directional topology into sink or source regions, and reported a previously unappreciated role of modulatory nuclei as critical effectors of inter-modular and network communicability. Finally, we demonstrated a close spatial correspondence between the mesoscale topography of the mouse connectome and its functional macroscale organization, showing that, like in primates and humans, the mouse cortical connectome is organized along two major topographical axes that can be linked to hierarchical patterns of laminar connectivity, and shape the topography of fMRI dynamic states, respectively. This investigation was paralleled by further studies aimed to more closely relate structural connectome features to the corresponding large scale functional networks of the mouse brain. We first focused on the mouse default mode network (DMN), describing its axonal substrates with sublaminar precision and cell-type specificity. We found that regions of the mouse DMN are predominantly located within the isocortex and exhibit preferential connectivity. Dedicated tract tracing experiments carried out by the Allen Brain Institute revealed that layer 2/3 DMN neurons projected mostly in the DMN, whereas layer 5 neurons project both in and out. Further analyses revealed the presence of separate in-DMN and out-DMN-projecting cell types with distinct genetic profiles. Lastly, we carried out a fine-grained comparison of functional topography and dynamic organization of large-scale fMRI networks in wakeful and anesthetized mice, relating the corresponding functional networks to the underlying architecture of structural connectivity. Recapitulating prior observations in conscious primates, we found that the awake mouse brain is subjected to a profound topological reconfiguration such to maximize cross-talk between cortical and subcortical neural systems, departing from the underlying structure of the axonal connectome. Taken together, these results advance our understanding of the foundational wiring principles of the mammalian connectome, and create opportunities for identifying targets of interventions to modulate brain function and its network structure in a physiologically-accessible species.

No description available.

This thesis is written as a single source introduction to Ramsey Theory for advanced undergraduates and graduate students. / Master of Science

Understanding higher level cognitive processes is a central problem in neuroscience. The Neuroidal model provides a useful framework for posing these problems in a computer science context. There has been significant recent work trying to understand memory capacity in the Neuroidal model but this work was done assuming that the network of neurons was an Erdos-Renyi random graph. However the network of neurons in the brain has been shown to exhibit small-world properties, which are not present in Erdos-Renyi graphs. In this research we explore replacing Erdos-Renyi graphs with Watts-Strogatz and Barabasi-Albert graphs in order to more accurately model the biological reality. We aim to investigate the implications for memory capacity and interference within the Neuroidal model.