Darren A. Narayan

Greedy rankings and arank numbers

A ranking on a graph is an assignment of positive integers to its vertices such that any path between two vertices of the same rank contains a vertex of strictly larger rank. A ranking is locally minimal if reducing the rank of any single vertex produces a non-ranking. A ranking is globally minimal if reducing the ranks of any set of vertices produces a non-ranking. A ranking is greedy if, for some ordering of the vertices, it is the ranking produced by assigning ranks in that order, always selecting the smallest possible rank. We will show that these three notions are equivalent. If a ranking satisfies one property it satisfies all three. As a consequence of this and known results on arank numbers of paths we improve known upper bounds for on-line ranking of paths and cycles.

Tiling Large Rectangles

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A comprehensive comparison of graph theory metrics for social networks

In this paper, we explore the relationship between two metrics that appear in the literature of social networks, local efficiency and the clustering coefficients. Next, we investigate these properties for a selection of real-world networks involving fMRI data from athletes and show for non-sparse graphs the relationship between the two properties is very close to linear.

Obtaining Funding and Support for Undergraduate Research.

Abstract Over the past decade there has been a dramatic increase in undergraduate research activities at colleges and universities nationwide. However, this comes at a time when budgets are being tightened and some institutions do not have the resources to pursue new initiatives. In this article we present some ideas for obtaining funding and support for building an undergraduate research program in mathematics.

Refining the clustering coefficient for analysis of social and neural network data

In this paper we show how a deeper analysis of the clustering coefficient in a network can be used to assess functional connections in the human brain. Our metric of edge clustering centrality considers the frequency at which an edge appears across all local subgraphs that are induced by each vertex and its neighbors. This analysis is tied to a problem from structural graph theory in which we seek the largest subgraph that is a Cartesian product of two complete bipartite graphs

A classification of tournaments having an acyclic tournament as a minimum feedback arc set

K_{1,m}

Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

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