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The main contribution of this paper is a six-step semi-automatic algorithm that obtains a recursion satisfied by a family of determinants by systematically and iteratively applying Laplace expansion to the underlying matrix family. The recursion allows explicit computation of the Binet form providing a closed formula for resistance distance between two specified nodes in a family of graphs. This approach is particularly suited for graph families with complex structures; the method is used to prove the 1 over 14 conjectured asymptotic formula for linear 3--trees. Additionally, although the literature on recursive formulas for resistance distances is quite large, the Fibonacci Quarterly and the Proceedings have almost no such results despite the fact that many recursions related to resistance distances involve the Fibonacci numbers. Therefore, a secondary purpose of the paper is to provide a brief introductory survey of graph families, accompanied by figures, Laplacian matrices, and typical recursive results, supported by a modest bibliography of current papers on many relevant graph families, in the hope to involve Fibonaccians in this active and beautiful field., Comment: To be presented, 24th International Fibonacci Conference, Claremont CA. 2024. Submitted to the Proceedings of the Fibonacci Association for Publication

The 2022 Fibonacci Conference held in Sarajevo introduced the Circuit Array, a two-dimensional array associated with the resistance labels of electrical circuits whose underlying graph when embedded in the Cartesian plane has the form of a triangular n-grid. The presentation used row reduction, a computational method alternative to the traditional method using the Laplacian. The presentation explored reconsidering the Circuit Array in terms of polynomials instead of numbers as a means to facilitate finding patterns. The present paper shows the fruitfulness of this approach. The main result of the current paper states that the characteristic polynomials corresponding to the recursions of single or multi-variable polynomial formulations of the Circuit Array exclusively have powers of 9 as roots. The proof methodology uses the approach of annihilators. Several initial cases and one major sub-case are proven., Comment: Still undergoing review by the Fibonacci Quarterly. This latest version incorporates suggestions by an anonymous referee and completely describes the factorizations of the factors of the numerator and denominator of the polynomial circuit array

Except for Koshy who devotes seven pages to applications of Fibonacci Numbers to electric circuits, most books and the Fibonacci Quarterly have been relatively silent on applications of graphs and electric circuits to Fibonacci numbers. This paper continues a recent trend of papers studying the interplay of graphs, circuits, and Fibonacci numbers by presenting and studying the Circuit Array, an infinite 2-dimensional array whose entries are electric resistances labelling edge values of circuits associated with a family of graphs. The Circuit Array has several features distinguishing it from other more familiar arrays such as the Binomial Array and Wythoff Array. For example, it can be proven modulo a strongly supported conjecture that the numerators of its left-most diagonal do not satisfy any linear, homogeneous, recursion, with constant coefficients (LHRCC). However, we conjecture with supporting numerical evidence an asymptotic formula involving $\pi$ satisfied by the left-most diagonal of the Circuit Array., Comment: Presented at Fibonacci Conference, Sarajevo, 2022

In [Evans, Francis 2022; Hendel] the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced $s$ times provided the original grid has at least $4s$ rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously., Comment: 12 pages

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between $0$ and $1$. Finally, we state several open conjectures that are motivated by empirical data.

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.

We examine whether social data can be used to predict how members of Major League Baseball (MLB) and members of the National Basketball Association (NBA) transition between teams during their career. We find that incorporating social data into various machine learning algorithms substantially improves the algorithms' ability to correctly determine these transitions. In particular, we measure how player performance, team fitness, and social data individually and collectively contribute to predicting these transitions. Incorporating individual performance and team fitness both improve the predictive accuracy of our algorithms. However, this improvement is dwarfed by the improvement seen when we include social data suggesting that social relationships have a comparatively large effect on player transitions in both MLB and in the NBA.

Comparative graph and network analysis play an important role in both systems biology and pattern recognition, but existing surveys on the topic have historically ignored or underserved one or the other of these fields. We present an integrative introduction to the key objectives and methods of graph and network comparison in each field, with the intent of remaining accessible to relative novices in order to mitigate the barrier to interdisciplinary idea crossover. To guide our investigation, and to quantitatively justify our assertions about what the key objectives and methods of each field are, we have constructed a citation network containing 5,793 vertices from the full reference lists of over two hundred relevant papers, which we collected by searching Google Scholar for ten different network comparison-related search terms. We investigate its basic statistics and community structure, and frame our presentation around the papers found to have high importance according to five different standard centrality measures.

A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood. For any connected graph $G$ with a 2-separator separating vertices $u$ and $v$, we show that the number of spanning trees and spanning 2-forests separating $u$ and $v$ can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if $u$ and $v$ are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.

In this paper, we introduce a three-dimensional mathematical model of collagen contraction with microbuckling based on the two-dimensional model previously developed by the authors. The model both qualitatively and quantitatively replicates experimental data including lattice contraction over a time course of 40 hours for lattices with various cell densities, cell density profiles within contracted lattices, radial cut angles in lattices, and cell force propagation within a lattice. The importance of the model lattice formation and the crucial nature of its connectivity are discussed including differences with models which do not include microbuckling. The model suggests that most cells within contracting lattices are engaged in directed motion.

We consider the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$. We call $G_n$ the straight linear 2-tree on $n$ vertices. Using $\Delta$--Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance $r_{G_n}(i,j)$ between any two vertices $i$ and $j$ of $G_n$. To our knowledge $\{G_n\}_{n=3}^\infty$ is the first nontrivial family with diameter going to $\infty$ for which all resistance distances have been explicitly calculated. Our result also gives formulae for the number of spanning trees and 2-forests in a straight linear 2-tree. We show that the maximal resistance distance in $G_n$ occurs between vertices 1 and $n$ and the minimal resistance distance occurs between vertices $n/2$ and $n/2+1$ for $n$ even (with a similar result for $n$ odd). It follows that $r_n(1,n) \to \infty$ as $n \to \infty$. Moreover, our explicit formula makes it possible to order the non-edges of $G_n$ exactly according to resistance distance, and this ordering agrees with the intuitive notion of distance on a graph. Consequently, $G_n$ is a geometric graph with entirely different properties than the random geometric graphs investigated in [6]. These results for straight linear 2-trees along with an example of a bent linear 2-tree and empirical results for additional graph classes convincingly demonstrate that resistance distance should not be discounted as a viable method for link prediction in geometric graphs.

In this note we consider the bent linear 2-tree and provide an explicit formula for the resistance distance $r_{G_n}(1,n)$ between the first and last vertices of the graph. We call the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$ a straight linear 2-tree. We define the graph $H_n$ with $V(H_n)= V(G_n)$ and $E(H_n) = (E(G_n) \cup \{k,k+3\})\setminus(\{k+1,k+3\}$ to be a bent linear 2-tree with bend at vertex $k$.

In this paper we present a method for knot insertion and degree elevation of generalized B-splines (GB-splines) via the local representation of these curves as piecewise functions. The use of local structures makes the refinement routines simpler to understand and implement.

In this paper we present a method for direct evaluation of generalized B-splines (GB-splines) via the local representation of these curves as piecewise functions. To accomplish this we introduce a local structure that makes GB-spline curves more amenable to the techniques used in constructing bases of higher degree. This basis is used to perform direct computation of piecewise representation of GB-spline bases and curves.