6 citations · 23 across the 28 of their papers we have counts for
30 papers · 1 filter
Spectral faux trees
Steve Butler, Elena D'Avanzo, Rachel Heikkinen +3
A spectral faux tree with respect to a given matrix is a graph which is not a tree but is cospectral with a tree for the given matrix. We consider the existence of spectral faux tr…
Complements of coalescing sets
Steve Butler, Elena D'Avanzo, Rachel Heikkinen +3
We consider matrices of the form , with being the diagonal matrix of degrees, being the adjacency matrix, and a fixed value. Given a graph and $B\subseteq V(G…
Hadamard diagonalizable graphs of order at most 36
Jane Breen, Steve Butler, Melissa Fuentes +7
If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries , then the matrix formed by taking the columns as the eigenvectors is a Hadamard mat…
Enumerating Parking Completions Using Join and Split
Ayomikun Adeniran, Steve Butler, Galen Dorpalen-Barry +5
Given a strictly increasing sequence with entries from , a parking completion is a sequence with and $|…
Spectral properties of the exponential distance matrix
Steve Butler, Elizabeth Coper, Aaron Li +2
Given a graph , the exponential distance matrix is defined entry-wise by letting the -entry be , where is the distance between th…
Properties of a -analogue of zero forcing
Steve Butler, Craig Erickson, Shaun Fallat +6
Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation…
Graphs with at most two trees in a forest building process
Steve Butler, Misa Hamanaka, Marie Hardt
Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The…
Graph switching, 2-ranks, and graphical Hadamard matrices
Aida Abiad, Steve Butler, Willem H. Haemers
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matri…
Ordered multiplicity inverse eigenvalue problem for graphs on six vertices
John Ahn, Christine Alar, Beth Bjorkman +6
For a graph , we associate a family of real symmetric matrices, , where for any , the location of the nonzero off-diagonal entries of a…
The inverse eigenvalue problem of a graph: Multiplicities and minors
Wayne Barrett, Steve Butler, Shaun M. Fallat +5
The inverse eigenvalue problem of a given graph is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G…
Enumerating multiplex juggling patterns
Steve Butler, Jeongyoon Choi, Kimyung Kim +1
Mathematics has been used in the exploration and enumeration of juggling patterns. In the case when we catch and throw one ball at a time the number of possible juggling patterns i…
A forest building process on simple graphs
Zhanar Berikkyzy, Steve Butler, Jay Cummings +4
Consider the following process on a simple graph without isolated vertices: Order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in…
Counting prime juggling patterns
Esther Banaian, Steve Butler, Christopher Cox +3
Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states t…
A generalization of Eulerian numbers via rook placements
Esther Banaian, Steve Butler, Christopher Cox +3
We consider a generalization of Eulerian numbers which count the number of placements of "rooks" on an board where there are exactly rooks in each row and each…
A cospectral family of graphs for the normalized Laplacian found by toggling
Steve Butler, Kristin Heysse
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix.…
Juggling card sequences
Steve Butler, Fan Chung, Jay Cummings +1
Juggling patterns can be described by a sequence of cards which keep track of the relative order of the balls at each step. This interpretation has many algebraic and combinatorial…
The mathematics of the flip and horseshoe shuffles
Steve Butler, Persi Diaconis, Ron Graham
We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is "reversed", and then the cards are interlaced. Flip shuffles are when the reversa…
Using twins and scaling to construct cospectral graphs for the normalized Laplacian
Steve Butler
The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges h…
Rainbow arithmetic progressions
Steve Butler, Craig Erickson, Leslie Hogben +8
In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers and , the expression $aw([n…
Zero forcing for inertia sets
Steve Butler, Jason Grout, H. Tracy Hall
Zero forcing is a combinatorial game played on a graph with a goal of turning all of the vertices of the graph black while having to use as few "unforced" moves as possible. This l…
Unrolling residues to avoid progressions
Steve Butler, Ron Graham, Linyuan Lu
We consider the problem of coloring with colors to minimize the number of monochromatic term arithmetic progressions (or -APs for short). We show how t…
Origami rings
Joe Buhler, Steve Butler, Warwick de Launey +1
Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let be the line in the complex plane through with…
A construction of cospectral graphs for the normalized Laplacian
Steve Butler, Jason Grout
We give a method to construct cospectral graphs for the normalized Laplacian by a local modification in some graphs with special structure. Namely, under some simple assumptions, w…
Hypercube orientations with only two in-degrees
Joe Buhler, Steve Butler, Ron Graham +1
We consider the problem of orienting the edges of the -dimensional hypercube so only two different in-degrees and occur. We show that this can be done, for two specified…