Steve Butler's Homepage
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400C Carver Hall

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Professor Butler with a beard

Videos available on YouTube

Mathematical interests
linear algebra
discrete mathematics
spectral graph theory

From February 2014 through May of 2015 I will be away from Iowa State University focusing on research. I can still be reached by email and will be happy to help you as best I can.

I am an Assistant Professor in the Department of Mathematics at Iowa State University (ISU). Prior to coming to ISU I did a three year NSF PostDoc under the supervision of Benny Sudakov at UCLA. Before that I earned my doctorate degree in mathematics at UC San Diego under the supervision of Fan Chung where I also worked extensively with Ron Graham.

From September 2012-February 2014, I maintained the Weekly Undergraduate Math Mailing (WUM^2) at Iowa State University. The archived announcements are available online.

A news story about by mathematics of juggling class I taught in the Fall of 2013.

Jump to Teaching, Publications, Miscellaneous--research, Talks/presentations, Links.

Teaching mathematics is one of the fun parts about my job. I have taught hundreds of students at four different universities.

Past courses:
Iowa State (2011-present)
  • Math 265: Calculus of several variables (Fall 2013)
  • Math 595: Mathematics of juggling (Fall 2013)
  • Math 607: Graph theory (Fall 2013)
  • Math 492: Undergraduate seminar (Spring 2013)
  • Math 314: Graph theory (Spring 2013)
  • Math 265: Calculus of several variables (Fall 2012)
  • Math 304: Combinatorics (Fall 2012)
  • Math 265: Calculus of several variables (Spring 2012)
  • Math 201: Introduction to proofs (Fall 2011)
  • Math 265: Calculus of several variables (Fall 2011)
  • Math 304: Combinatorics (Fall 2011 -- independent study)
  • UCLA (2008-2010)
  • Math 32A: Calculus of several variables
  • Math 31A: Differential and Integral Calculus (videos of lectures)
  • Math 180: Combinatorics (lecture notes)
  • Math 3B: Calculus for life science students
  • Math 61: Introduction to discrete structures
  • UCSD (2007)
  • Math 10A: Differential calculus
  • BYU (2001-2003)
  • Differential calculus
  • Integral calculus
  • Trigonometry (5 classes) (lecture notes)

  • Graduate student:
  • Steven Osborne (joint student with Leslie Hogben); Ph.D. in Fall 2013.

    Papers that have been submitted:

  • B. Sandlund, K. Staley, M. Dixon and S. Butler, Numerical Tic-Tac-Toe on the 4x4 Board.
  • W. Barrett, S. Butler, M. Catral, S. Fallat, H. T. Hall, L. Hogben and M. Young, The maximum nullity of a complete subdivision graph is equal to its zero forcing number.
  • W. Barrett, S. Butler, M. Catral, S. Fallat, H. T. Hall, L. Hogben, P. van den Driessche and M. Young, The principal rank characteristic sequence over various fields.
  • S. Butler, M. Catral, S. Fallat, H. T. Hall, L. Hogben, P. van den Driessche and M. Young, The enhanced principal rank characteristic sequence.
  • S. Butler, J. Grout and H. T. Hall, Using variants of zero forcing to bound the inertia set of a graph.

    Papers that will appear:

  • S. Butler, R. Graham and L. Lu, Unrolling residues to avoid progressions, to appear in Mathematics Magazine.
  • S. Butler and F. Chung, Spectral Graph Theory, to appear in Handbook of Linear Algebra, Second Edition, CCR Press.
  • S. Butler, R. Graham and R. Stong, Collapsing numbers in bases 2, 3, and beyond, to appear in the proceedings of Gathering for Gardner X.
  • S. Butler and S. Osborne, Counting tilings by taking walks, to appear in Journal of Combinatorial Mathematics and Combinatorial Computing 88 (2014), ???-???.
  • S. Butler, Backwards addition, to appear in the proceedings of Gathering for Gardner X.
  • A. Alhakim, S. Butler and R. Graham, de Bruijn sequences with varying combs, to appear in INTEGERS: The Electronic Journal of Combinatorial Number Theory.

    Papers that have appeared:

    1. J. Bodeen, S. Butler, T. Kim, X. Sun and S. Wang, Tiling a strip with triangles, Electronic Journal of Combinatorics 21 (2014), P1.7, 15pp.
    2. S. Butler, R. Graham, G. Guettler and C. Mallows, An interstice relationship for flowers with four petals, Journal of Geometry 104 (2013), 421-438.
    3. S. Butler and M. Young, Throttling zero forcing propagation speed, Australasian Journal of Combinatorics 57 (2013), 65-71.
    4. S. Butler, E. Demaine, R. Graham and T. Tachi, Constructing points through folding and intersection, International Journal of Computational Geometry & Applications 23 (2013), 49-64.
    5. S. Butler and R. Graham, Subdivision using angle bisectors is dense in the space of triangles, The American Mathematical Monthly 120 (2013), 622-630.
    6. J. Buhler, S. Butler, W. de Launey and R. Graham, Origami rings, Journal of the Australian Mathematical Society 92 (2012), 299-311.
    7. S. Butler and R. Graham, A note on marking lines in [k]n, Designs, Codes and Cryptography 65 (2012), 165-175.
    8. W. Barrett, S. Butler, H. T. Hall, J. Sinkovic, W. So, C. Starr and A. Yielding, Computing inertia sets using atoms, Linear Algebra and its Applications 436 (2012), 4489-4502.
    9. S. Butler and J. Grout, A construction of cospectral graphs for the normalized Laplacian, Electronic Journal of Combinatorics 18 (2011), #231, 20pp.
    10. J. Buhler, S. Butler, R. Graham and E. Tressler, Hypercube orientations with only two in-degrees, Journal of Combinatorial Theory, Series A 118 (2011), 1695-1702.
    11. S. Butler, K. Costello and R. Graham, Finding patterns avoiding many monochromatic constellations, Experimental Mathematics 19 (2010), number 4, 399-411.
    12. S. Butler and R. Graham, Iterated triangle partitions, Fete of Combinatorics and Computer Science, G. Katona, A. Schrijver, T. Szonyi, eds., Bolyai Society Mathematical Studies 29, Springer-Verlag, Heidelberg (2010), 23-42.
    13. S. Butler, F. Chung, R. Graham and M. Laczkovich, Tiling polygons with lattice triangles, Discrete & Computational Geometry 44 (2010), 896-903.
    14. S. Butler, R. Graham, G. Guettler and C. Mallows, Irreducible Apollonian configurations and packings, Discrete & Computational Geometry 44 (2010), 487-506.
    15. S. Butler, Eigenvalues of 2-edge-coverings, Linear and Multilinear Algebra 58 (2010), 413-423.
    16. S. Butler and R. Graham, Shuffling with ordered cards, Journal of Combinatorics 1 (2010), 121-139.
    17. S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, Annals of Combinatorics 13 (2010), 413-424.
    18. S. Butler and F. Chung, Small spectral gap in the combinatorial Laplacian implies Hamiltonian, Annals of Combinatorics 13 (2010), 403-412.
    19. S. Butler, P. Horn and E. Tressler, Intersecting domino tilings, The Fibonacci Quarterly 48 (2010), 114-120.
    20. S. Butler, R. Graham and J. Mao, How to play the majority game with a liar, Discrete Mathematics 310 (6 February 2010), 622-629.
    21. S. Butler and P. Karasik, A note on nested sums, Journal of Integer Sequences 13 (2010), article 10.4.4, 8 pp.
    22. S. Butler, Cospectral graphs for both the adjacency and normalized Laplacian Matrices, Linear and Multilinear Algebra 58 (2010), 387-390.
    23. S. Butler, Induced-universal graphs for graphs with bounded maximum degree, Graphs and Combinatorics 25 (2009), 461-468.
    24. S. Butler, The lost daughters of Gergonne, Forum Geometricorum 9 (2009), 19-26.
    25. S. Butler, R. Graham and N. Zang, Optimal jumping patterns, Journal of Combinatorics and Number Theory 1 (2009), 1-13.
    26. S. Butler, M. T. Hajiaghayi, R. D. Kleinberg and T. Leighton, Hat guessing games, SIAM Journal on Discrete Mathematics 22 (2008), 592-605. Republished in SIAM Review 51 (2009), 399-413.
    27. S. Butler, R. Graham and N. Zang, Jumping sequences, Journal of Integer Sequences 11 (2008), 13 pp.
    28. S. Butler, Estimating the number of graphs containing very long induced paths, Ars Combinatoria 88 (2008), 321-332.
    29. AIM minimum rank-special graphs work group, Zero forcing sets and the minimum rank of graphs, Linear Algebra and its Applications 428 (2008), 1628-1648.
    30. M. Bousquet-Melou and S. Butler, Forest-like permutations, Annals of Combinatorics 11 (2007), 335-354.
    31. S. Butler, R. Graham and J. Mao, How to play the majority game with liars, AAIM 2007, Lecture Notes in Computer Science 4508, Springer-Verlag, 221-230.
    32. S. Butler, Interlacing for weighted graphs using the normalized Laplacian, Electronic Journal of Linear Algebra 16 (2007), 90-98.
    33. S. Butler, Using discrepancy to control singular values for nonnegative matrices, Linear Algebra and its Applications 419 (2006), 486-493.
    34. S. Butler, 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.
    35. S. Butler, Tangent line transformations, The College Mathematics Journal 34 (2003), 105-106.

    Coauthors: AIM minimum rank-special graphs work group, Wayne Barrett, Mireille Bousquet-Melou, Joe Buhler, Fan Chung, Kevin Costello, Warwick de Launey, Erik Demaine, Ron Graham, Jason Grout, Gerhard Guettler, Mohammad Hajiaghayi, H. Tracy Hall, Paul Horn, Pavel Karasik, Robert Kleinberg, Miklos Laczkovich, Tom Leighton, Colin Mallows, Jia Mao, Steven Osborne, John Sinkovic, Wasin So, Colin Starr, Richard Stong, Tomohiro Tachi, Eric Tressler, Amy Yielding, Nan Zang


    Eigenvalues and structures of graphs -- Ph.D. dissertation written at UC San Diego under the direction of Fan Chung. Much of this has shown up in various papers I have published.

    Bounding the number of graphs containing very long induced paths -- Masters thesis written at Brigham Young University under the direction of Wayne Barrett.

    JAVA applets written to look at what happens to daughters when bisecting triangles using some well defined central point. This is related to the papers Iterated triangle partitions and The lost daughters of Gergonne.

    The papers have generated many beautiful images and some can be found in the blog of Loic Seguin-Charbonneau "Triangulations creatives".

    For the paper Finding patterns avoiding many monochromatic constellations we generated a Maple worksheet that gives a method to compute the coefficient of a block pattern given a constellation. We also have an extended Maple worksheet that contains more information about good block colorings. There is also an interactive JAVA applet that allows you to see what happens to the coefficient of a pattern as we perturb the pattern and/or the constellation. In addition we have the JAVA programs used to generate block patterns for 4-APs as well as the general pattern [0,q,1].

    For the paper Irreducible Apollonian configurations and packings we generated a Maple worksheet that automates finding the center and radii of a packing placed in standard position. We also have a list of all irreducible configuration of circles with between 10 and 15 circles that can be used in the worksheet. Finally, there is also a collection of packings with various curvatures marked in the circles.

    While visiting The Center for Combinatorics at Nankai University in September 2006 I gave three lectures that served as an introduction to the field of graph theory.
  • The first lecture was an introduction to three common spectra that looked at the adjacency matrix, the Laplacian matrix, and the normalized Laplacian and why each one might be used.
  • The second lecture was about applications of Courant-Fischer and gave some basic applications including examining what happens with the removal of a subgraph.
  • The third lecture discussed Cheeger constants and discrepancy and looked at various properties of edge expansion in graphs.

  • I have posed various problems which have appeared. These include American Mathematical Monthly #11030, #11265 and Mathematics Magazine #1668, #1730, #1761, #Q1008.

    In Fall of 2004 Fan Chung taught a course about the combinatorics of patterns in subsets and graphs. I worked to collect and collate the notes for the course into a PDF document; of course Fan did all the hard work of teaching the course.

    A collection of the simplest aperiodic strongly connected directed graphs and their spectrums under the definition of the normalized Laplacian (as defined by Chung).

    Miscellaneous unpublished notes:
  • Relating the arboricity with the chromatic number of a graph
  • Determining the underlying functions for Cauchy power and exponential forms
  • A property of the gcd; and The Moebius transform of the triangular numbers
  • The art of juggling with two balls
  • A property of positive semidefinite matrices
  • Tangent line transformations
  • Method for doing the (approximate) Markov chains for bisection

  • A letter to the editor of my old school about the slogan do the math used for a fundraising campaign.


  • Finding patterns avoiding many monochromatic constellations (slides), delivered at UCSD combinatorics seminar; UCLA combinatorics seminar; AMS 2009 Fall Western Section Meeting; AMS 2010 Spring Central Section Meeting; Colloquim at Michigan Technological University; AMS 2011 Fall Southeastern Section Meeting.
  • Applications and limitations for the normalized Laplacian, delivered at 2012 Joint Mathematics Meeting.
  • Cospectral graphs for the normalized Laplacian (slides), delivered at 2010 SIAM Conference on Discrete Mathematics, ISU Discrete Mathematics Seminar.
  • Hat games and hypercube orientations (slides), delivered at Discrete Mathematics Seminar at Iowa State University; Drake University; Combinatorics Seminar at UC San Diego.
  • Shuffling with ordered cards (slides), delivered at Combinatorics, Groups, Algorithms, and Complexity: Conference in honor of Laci Babai's 60th birthday; 2010 Western Number Theory Conference.
  • Iterated partitions of triangles, delivered at The 20th Canadian Conference on Computational Geometry (co-presenter with Ron Graham); The Mathematical Interests of Peter Borwein (co-presenter with Ron Graham); Drake University.
  • Tiling polygons with lattice triangles (slides), delivered at INTEGERS Conference 2009.
  • Jumping sequences (slides), delivered at UCSD combinatorics seminar; UCLA combinatorics seminar.
  • Induced universal graphs (slides), delivered at UCSD combinatorics seminar; SDSU REU; BYU colloquium; Sixth Czech-Slovak International Symposium; AMS 2007 Western Sectional Meeting; ISU Discrete Mathematics Seminar; MIGHTY LII.
  • Eigenvalues of 2-edge-coverings, delivered at CMS Winter 2007 Meeting; UCSD combinatorics seminar.
  • An Erdos-Ko-Rado problem on the strip (slides), delivered at GSCC 2008.
  • How to play the majority game with liars (slides), delivered at AAIM07.
  • Enumerating (multiplex) juggling sequences, delivered at Horizon of Combinatorics; UCSD combinatorics seminar.
  • On permutations which are 1324 and (bar 2143) avoiding, delivered at UC Berkeley combinatorics seminar; UCSD combinatorics seminar; CalTech combinatorics seminar.
  • Fibonacci meets Erdos-Ko-Rado (slides), delivered at ISU math club; Drake math club.
  • Generalizations of Apollonian circle packings, delivered at BYU Colloquium; ISU Colloquium.
  • Bounding inertia sets of a graph, delivered at AMS 2011 Central Sectional meeting; 2012 Cumberland Combinatorics Conference.


  • Ryan Martin
  • Michael Young
  • Leslie Hogben
  • Maria Axenovich

  • Ron Graham
  • Fan Chung
  • Jacques Verstraete
  • Jeff Remmel
  • Adriano Garsia
  • Benny Sudakov
  • Igor Pak
  • Terry Tao

  • Noga Alon
  • Art Benjamin
  • Mirielle Bousquet-Melou
  • Andries Brouwer
  • Richard Brualdi
  • Peter Cameron
  • Bill Chen
  • Sebi Cioaba
  • Josh Cooper
  • Rob Ellis
  • David Eppstein
  • Jacob Fox
  • Zoltan Furedi
  • Chris Godsil
  • Willem Haemers
  • Paul Horn
  • Steve Kirkland
  • Don Knuth
  • Andre Kundgen
  • Felix Lazebnik
  • Po-Shen Loh
  • Lincoln Lu
  • Bill Martin
  • Bojan Mohar
  • Joel Spencer
  • William Trotter
  • Doug West
  • Journals

  • Electronic J. of Combinatorics
  • J. of Combinatorics
  • Discrete Mathematics
  • European J. of Combinatorics
  • J. of Combinatorial Theory, Series A
  • Annals of Combinatorics
  • Combinatorics, Probability and Computing
  • Combinatorica
  • Graphs and Combinatorics
  • J. of Graph Theory
  • Discrete & Computational Geometry
  • SIAM J. on Discrete Mathematics
  • Random Structures & Algorithms
  • Australasian J. of Combinatorics
  • Contributions to Discrete Mathematics
  • Fibonacci Quarterly
  • Linear Algebra and its Applications
  • Electronic J. of Linear Algebra
  • SIAM J. on Matrix Analysis and Applications
  • Mathematical Intelligencer
  • Blogs/social media

  • Terry Tao blog
  • Tim Gowers blog
  • David Bressoud
  • Math Overflow
  • Reddit -- math

  • TeX Users Group (TUG)
  • TeX stackexchange
  • TeXShop (for OS X)

  • SAGE
  • Online sage server
  • ISU Sage server
  • Python programming

  • Math conferences
  • OEIS
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  • Last modified: 7 February 2014