Publications of Steve Butler
Ordered multiplicity inverse eigenvalue problem for graphs on six vertices, (with John Ahn, Christine Alar, Beth Bjorkman, Joshua Carlson, Audrey Goodnight, Haley Knox, Casandra Monroe, and Michael Wigal), to appear in Electronic Journal of Linear Algebra.
Paperclip graphs, (with Erik D. Demaine, Martin L. Demaine, Ron Graham, Adam Hesterberg, Jason Ku, Jayson Lynch, and Tadashi Tokieda), submitted.
Properties of a q-analogue of zero forcing, (with Craig Erickson, Shaun Fallat, H. Tracy Hall, Brenda Kroschel, Jephian C.-H. Lin, Bryan Shader, Nathan Warnberg, and Boting Yang), to appear in Graphs and Combinatorics.
Spectral properties of the exponential distance matrix, (with Elizabeth Cooper, Aaron Li, Kate Lorenzen, Zoë Schopick), submitted.
Hadamard diagonalizable graphs of order at most 36 (with Jane Breen, Melissa Fuentes, Bernard Lidický, Michael Phillips, Alexander W. N. Riasanovksy, Sung-Yell Song, Ralihe R. Villagrán, Cedar Wiseman, and Xiaohong Zhang), submitted
The mathematical life of Fan Chung, AMS Notices 67 (2020), 327-335.
The inverse eigenvalue problem of a graph: Multiplicities and minors, (with Wayne Barrett, Shaun Fallat, Tracy Hall, Leslie Hogben, Jephian Lin, Bryan Shader, and Michael Young), JCTB 142 (2020), 276-306. [doi:10.1016/j.jctb.2019.10.005]
A jaunt in spectral graph theory, 50 Years of Combinatorics, Graph Theory, and Computing, F. Chung, R. Graham, F. Hoffman, L. Hogben, R. Mullin, D. West, eds. CRC Press, 2020, 213-238.
Counting tilings by taking walks in a graph, (with Jason Ekstrand and Stephen Osborne), A Project-Based Guide to Undergraduate Research in Mathematics, P. E. Harris, E. Insko, A. Wootton, eds. Springer, 2020, 153-176.
Enumerating parking completions using Join and Split, (with Ayomikun Adeniran, Galen Dorpalen-Barry, Pamela E. Harris, Cyrus Hettle, Qingzhong Liang, Jeremy L. Martin, Hayan Nam), Electronic Journal of Combinatorics 27 (2020), #P2.44, 19 pps. [doi:10.37236/9194]
Enumerating multiplex juggling patterns, (with Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo), Journal of Integer Sequences 22 (2019), 19.1.7., 21 pp.
Proof without words: Independent sets in grid graphs and tilings of Aztec diamonds, Mathematics Magazine 92 (2019), 126-127. [doi:10.1080/0025570X.2019.1542233]
Graphs with at most two trees in a forest building process, (with Misa Hamanaka and Marie Hardt), Involve 12 (2019), 659-670. [doi:10.2140/involve.2019.12.659]
On the genus of a quotient of a numerical semigroup, (with Ayomikun Adeniran, Colin Defant, Yibo Gao, Pamela E. Harris, Cyrus Hettle, Qingzhong Liang, Hayan Nam, Adam Volk), Semigroup Forum 98 (2019) 690-700. [doi:10.1007/s00233-018-9989-3]
Graph switching, 2-ranks, and graphical Hadamard matrices, (with Aida Abiad and Willem Haemers),Discrete Mathematics 342 (2019), 2850-2855. [doi:10.1016/j.disc.2018.11.022].
Computing Kemeny's constant for a barbell graph, (with Jane Breen, Nicklas Day, Colt DeArmond, Kate Lorenzen, Haoyang Qian, Jacob Riesen), Electronic Journal of Linear Algebra 35 (2019), 583-598. [doi:10.13001/1081-3810.4095]
A forest building process on simple graphs, (with Zhanar Berikkyzy, Jay Cummings, Kristin Heysse, Paul Horn, Ruth Luo, and Brent Moran), Discrete Mathematics 341 (2018), 497-507. [doi:10.1016/j.disc.2017.09.017]
Apollonian ring packings, (with Adrian Bolt, and Espen Hovland), in Connections in Discrete Mathematics, S. Butler, J. Cooper, G. Hurlbert, eds. Cambridge University Press, 283-296.
A generalization of Eulerian numbers via rook placements, (with Esther Banaian, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce), Involve 10 (2017), 691-705. [doi:10.2140/involve.2017.10.691]
The enhanced principal rank characteristic sequence for Hermitian matrices, (with Minerva Catral, Tracy Hall, Leslie Hogben, Xavier Martinez-Rivera, Bryan Shader, and Pauline van den Driessche), Electronic Journal of Linear Algebra 32 (2017), 58-75. [doi:10.13001/1081-3810.3249]
Zero forcing propagation time on oriented graphs, (with Adam Berliner, Chassidy Bozeman, Minerva Catral, Leslie Hogben, Brenda Kroschel, Jephian Chin-Hung Lin, Nathan Warnberg, and Michael Young), Discrete Applied Mathematics 224 (2017), 45-59. [doi:10.1016/j.dam.2017.02.017]
Symmetric 0-1 matrices with inverses having two distinct values and constant diagonal, (with Wayne Barrett and Tracy Hall), Linear Algebra and Its Applications 491 (2016), 41-55. [doi:10.1016/j.laa.2015.02.019]
The enhanced principal rank characteristic sequence, (with Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, Pauline van den Driessche, and Michael Young), Linear Algebra and Its Applications 498 (2016), 181-200. [doi:10.1016/j.laa.2015.03.023]
Algebraic aspects of the normalized Laplacian, in Recent Trends in Combinatorics, Andrew Beveridge, Jerrold Griggs, Leslie Hogben, Gregg Musiker, and Prasad Tetali, eds., Springer, 2016, 295-315.
The mathematics of the flip and horseshoe shuffles, (with Persi Diaconis and Ron Graham), American Mathematical Monthly 123 (2016), 542-556. [doi:10.4169/amer.math.monthly.123.6.542]
A cospectral family of graphs for the normalized Laplacian found by toggling, (with Kristin Heysse), Linear Algebra and its Applications 507 (2016), 499-512. [doi:10.1016/j.laa.2016.06.033]
Rainbow arithmetic progressions, (with Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard L. Kramer, Jephian Chin-Hung Lin, Ryan R. Martin, Derrick Stolee, Nathan Warnberg, and Michael Young), Journal of Combinatorics 7 (2016), 595-626. [doi:10.4310/JOC.2016.v7.n4.a3]
Counting prime juggling patterns, (with Esther Banaian, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce), Graphs and Combinatorics 32 (2016), 1675-1688. [doi:10.1007/s00373-016-1711-1].
Using twins and scaling to construct cospectral graphs for the normalized Laplacian, Electronic Journal of Linear Algebra 28 (2015), 54-68. [doi:10.13001/1081-3810.2989]
Using variants of zero forcing to bound the inertia set of a graph, (with Jason Grout and Tracy Hall), Electronic Journal of Linear Algebra 30 (2015), 1-18. [doi:10.13001/1081-3810.2900]
Egyptian fractions with each denominator having three distinct prime divisors, (with Paul Erdos and Ron Graham), INTEGERS: The Electronic Journal of Combinatorial Number Theory 15 (2015) A51, 9 pp.
Tiling a strip with triangles, (with John Bodeen, Taekyoung Kim, Xiyuan Sun, and Shenzhi Wang), Electronic Journal of Combinatorics 21 (2014), P1.7, 15pp.
Counting tilings by taking walks, (with Steven Osborne), The Journal of Combinatorial Mathematics and Combinatorial Computing 88 (2014), 83-94.
The maximum nullity of a complete subdivision graph is equal to its zero forcing number, (with Wayne Barrett, Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, and Michael Young), The Electronic Journal of Linear Algebra 27 (2014), 444-457. [doi:10.13001/1081-3810.1629]
The principal rank characteristic sequence over various fields, (with Wayne Barrett, Minnie Catral, Shaun Fallat, Tracy Hall, Leslie Hogben, Pauline van den Driessche, and Michael Young), Linear Algebra and Its Applications 459 (2014), 222-236. [doi:10.1016/j.laa.2014.06.045]
de Bruijn sequences with varying combs, (with Abbas Alhakim and Ron Graham), INTEGERS: The Electronic Journal of Combinatorial Number Theory 14A (2014) A1, 23 pp.
Spectral graph theory, (with Fan Chung), in Handbook of Linear Algebra, second edition, Leslie Hogben ed., 14pp.
A gentle introduction to the normalized Laplacian, IMAGE 53 (Fall 2014), 19-27.
An interstice relationship for flowers with four petals, (with Ron Graham, Gerhard Guettler, and Colin Mallows), Journal of Geometry 104 (2013), 421-438. [doi:10.1007/s00022-013-0173-3]
Throttling zero forcing propagation speed on graphs, (with Michael Young), Australasian Journal of Combinatorics 57 (2013), 65-71.
Constructing points through folding and intersection, (with Erik Demaine, Ron Graham, and Tomohiro Tachi), International Journal of Computational Geometry & Applications 23 (2013), 49-64. [doi:10.1142/S0218195913500039]
Subdivision using angle bisectors is dense in the space of triangles, (with Ron Graham), The American Mathematical Monthly 120 (2013), 622-630. [doi:10.4169/amer.math.monthly.120.07.622]
Computing inertia sets using atoms, (with Wayne Barrett, Tracy Hall, John Sinkovic, Wasin So, Colin Starr, and Amy Yielding), Linear Algebra and its Applications 436 (2012), 4489-4502. [doi:10.1016/j.laa.2011.08.026]
A construction of cospectral graphs for the normalized Laplacian, (with Jason Grout) Electronic Journal of Combinatorics 18 (2011), #231, 20pp.
Finding patterns avoiding many monochromatic constellations, (with Kevin Costello and Ron Graham), Experimental Mathematics 19 (2010), number 4, 399-411. [doi:10.1080/10586458.2010.10390631]
Iterated triangle partitions, (with Ron Graham), Fete of Combinatorics and Computer Science, G. Katona, A. Schrijver, T. Szonyi, eds., Bolyai Society Mathematical Studies 29, Springer-Verlag, Heidelberg (2010), 23-42. [doi:10.1007/978-3-642-13580-4_2]
Irreducible Apollonian configurations and packings, (with Ron Graham, Gerhard Guettler, and Colin Mallows), Discrete & Computational Geometry 44 (2010), 487-506. [doi:10.1007/s00454-009-9216-9]
Small spectral gap in the combinatorial Laplacian implies Hamiltonian, (with Fan Chung), Annals of Combinatorics 13 (2010), 403-412. [doi:10.1007/s00026-009-0039-4]
Intersecting domino tilings, (with Paul Horn and Eric Tressler), The Fibonacci Quarterly 48 (2010), 114-120.
A note on nested sums, (with Pavel Karasik), Journal of Integer Sequences 13 (2010), article 10.4.4, 8 pp.
A note about cospectral graphs for the adjacency and normalized Laplacian matrices, Linear and Multilinear Algebra 58 (2010), 387-390. [doi:10.1080/03081080902722741]
Induced-universal graphs for graphs with bounded maximum degree, Graphs and Combinatorics 25 (2009), 461-468. [doi:10.1007/s00373-009-0860-x]
The lost daughters of Gergonne, Forum Geometricorum 9 (2009), 19-26.
Optimal jumping patterns, (with Ron Graham and Nan Zang), Journal of Combinatorics and Number Theory 1 (2009), 1-13.
Hat guessing games, (with Mohammad Hajiaghayi, Robert Kleinberg, and Tom Leighton), SIAM Journal on Discrete Mathematics 22 (2008), 592-605. [doi:10.1137/060652774]; Republished in SIAM Review 51 (2009), 399-413. [doi:10.1137/080743470]
Jumping sequences, (with Ron Graham and Nan Zang), Journal of Integer Sequences 11 (2008), 13 pp.
Estimating the number of graphs containing very long induced paths, Ars Combinatoria 88 (2008), 321-332.
Interlacing for weighted graphs using the normalized Laplacian, Electronic Journal of Linear Algebra 16 (2007), 90-98. [doi:10.13001/1081-3810.1185]
Relating singular values and discrepancy of weighted directed graphs, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm (Miami, FL, 2006), 1112-1116. [doi:10.1145/1109557.1109680]
Using discrepancy to control singular values for nonnegative matrices, Linear Algebra and its Applications 419 (2006), 486-493. [doi:10.1016/j.laa.2006.05.015]
Miscellaneous writings and musings
Eigenvalues and structures of graphs, Ph.D dissertation, written under the direction of Fan Chung.
Bounding the number of graphs containing very long induced paths, MS Thesis, written under the direction of Wayne Barrett
Determining the underlying functions for Cauchy power and exponential forms, research done as an undergraduate student
Tangent line transformations, research done as an undergraduate student
Posed problems (Monthly #11030, #11265; Math Magazine #1668, #1730, #1761, #Q1008)
Memories about Ron Graham
In September 2006 I visited the Center for Combinatorics at Nankai University and gave a series of three lectures. The notes for these lectures are given here.
In April 2011 I gave a guest lecture about cospectral graphs at UCSD. The notes for this lecture is given here.
Some proofs without words (or not very many).
Various bits and pieces.
Contributions to the Gathering for Gardner over the years.
Letter to the editor of school newspaper written as an undergraduate.