An Upper Bound for the Representation Number of Graphs with Fixed Order

A graph has a representation modulo n if there exists an injective map f: {V(G)} -\u3e {0, 1, ... , n @ 1} such that vertices u and v are adjacent if and only if |f(u)@f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erdos and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible (Refer to PDF file for exact formulas)

A Classification of Tournaments Having an Acyclic Tournament as a Minimum Feedback Arc Set

Given a tournament with an acyclic tournament as a feedback arc set we give necessary and sufficient conditions for this feedback arc set to have minimum size

Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

There are many known connections between determinants of tridiagonal matrices and the Fibonacci and Lucas numbers

Minimal k-rankings and the a-rank number of a path

Given a graph G, a function f: V(G) -\u3e {1, 2, ..., k} is a k-ranking of G if f(u) = f(v) implies every u - v path contains a vertex w such that f(w) \u3e f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The a-rank number of G, denoted u,(G) equals the largest k such that G has a minimal k-ranking. We establish new results involving minimal rankings of paths and in particular we determine u(Pn), a problem suggested by Laskar and Pillone in 2000. We show u(Pn) = [log2 (n + 1)] + [log2(n + 1 - (2^([log2n]-1))] (Refer to PDF file for exact formulas)

Fibonacci determinants

Fibonacci numbers don\u27t occur everywhere but they can arise in unexpected places, such as Hessenberg matrices. (You don\u27t know what a Hessenberg matrix is? Better find out!

Tournaments with a Transitive Subtournament as a Feedback Arc Set

Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph is defined to be r(D) = |V (T)|−|V (D)| . The case where D is a tournament Tn was studied by Isaak in 1995 using an integer linear programming formulation. In particular, this approach was used to produce lower bounds for r(Tn), and it was conjectured that the given bounds were tight. We examine the class of tournaments where n = 2k +2k−2 and show the known lower bounds for r(Tn) are best possible

Minimal Rankings and the A-Rank Number of a Path

Given a graph G, a function f:V(G)→ {1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)\u3ef(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψr(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψr(Pn), a problem posed by Laskar and Pillone in 2000 (Refer to PDF file for exact formulas)

Characterizing Edge Betweenness-Uniform graphs

The \emph{betweenness centality} of an edge $e$ is, summed over all $u,v\in V(G)$, the ratio of the number of shortest $u,v$-paths in $G$ containing $e$ to the number of shortest $u,v$-paths in $G$. Graphs whose vertices all have the same edge betweenness centrality are called \emph{edge betweeness-uniform}. It was recently shown by Madaras, Hurajová, Newman, Miranda, Fl\\u27{o}rez, and Narayan that of the over 11.7 million graphs with ten vertices or fewer, only four graphs are edge betweenness-uniform but not edge-transitive.In this paper we present new results involving properties of betweenness-uniform graphs

Characterization of Graphs With Failed Skew Zero Forcing Number of 1

Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. Hence the failed zero forcing number of a graph was defined to be the size of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added to $S$ in the next iteration. The difference is that vertices that are not in $S$ can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by $F^{-}(G)$. In this paper we provide a complete characterization of all graphs with $F^{-}(G)=1$. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of $1$ are either: the union of two isolated vertices; $P_3$; $K_3$; or $K_4$. In this paper we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with $F^{-}(G)=1$.Comment: 8 pages, this research was supported by the National Science Foundation Research forUndergraduates Award 195018