Mike Krebs

Teaching

• Class websites (Requires permission to access.  I'm happy to share with other instructors, students, etc.  Please email me at [email protected] to ask for access.  Please explain who you are and why you'd like access, and please have a verifiable email address, ideally with a .edu domain.)
• Welcome to the world of mathematics: An introduction to being a math major (intro to a set of course notes for a proofs class)

Publications

• Chromatic numbers of Cayley graphs of abelian groups: Cases of small dimension and rank, joint work with Cal State LA student J. Cervantes.  Quaestiones Mathematicae, 21 Jul. 2026, pp. 1–24.
• Authors' Notes for "Chromatic numbers of abelian Cayley graphs: A matrix method"  (bonus content: a Mathematica file with code for computing the chromatic number of a standardized abelian Cayley graph associated to a 1 x r, m x 1, 2 2 x 2, or 3 x 2 matrix.)
• Chromatic numbers of circulants with indispensable generators, joint work with Ferdous Ahmed (UCI student), David Asraf (UCSD student), David Bonds (CSULA student), Jonathan Davidson (CSULA student), Yunhee Jang (Rice University student), Anand Prakash (CSULA student), and Edgar Yak-De Padua (CSULA student).  2026.  Preprint.
• Authors’ notes for “Chromatic numbers of circulants with indispensable generators” by Ferdous Ahmed, David Asraf, David Bonds, Jonathan Davidson, Yunhee Jang, Mike Krebs, Anand Prakash, and Edgar Yak-De Padua.
• Brian Chung, Mike Krebs.  Dimensions of Mycielskians of cycles.  2026.  Preprint.
• Mike Krebs, Maya Sankar.  Abelian groups without 3-chromatic Cayley graphs.  Journal of Combinatorial Theory, Series B, Volume 179, July 2026, Pages 205-218
• On Matsushita $\pi_1^2$ discrete fundamental groups, joint work with Cal State LA students A. Pan, A. Prakash.  2025.  Preprint.
• Four plane unit vectors generate a 3-colorable graph, joint work with Cal State LA students K. Eng, T. Harris, M. Meeks, and C. Schmidt.  2025.  Preprint.
• Mike Krebs, Alejandro Leyva.  Chromatic numbers of rank-two Abelian Cayley graphs.  2025.  Preprint.
• A Graph Theorist Plants a Tree, The College Mathematics Journal (2024), joint work with Cal State LA students A. Carrillo, J. Cervantes, and F. Leon.  This paper was awarded a 2026 George Pólya Award for Expository Writing by the MAA.
• Chromatic numbers of Cayley graphs of abelian groups: A matrix method, joint work with Cal State LA student J. Cervantes.  Linear Algebra and its Applications, Volume 676, 1 November 2023, pp. 277-295.  (The arXiv version this links to corrects an error in an example that appears in the published version.)
• Kazhdan constants and isomorphic graph pairs, joint work with Cal State LA students M. Davila, T. Hayes, M. Krebs, and M. Reyes.  Algebra and Discrete Mathematics, Volume 35 (2023). Number 1, pp. 73-85.
• Clique numbers of finite unit-quadrance graphs, Journal of Algebraic Combinatorics, 57, 1-20 (2023)
• What Baseball Team Does PageRank Say Was The Best Ever?, Baseball Research Journal, Spring 2022.  Joint work with Cal State LA student Gwen Ostergren
• Finite epsilon unit-distance graphs.  Journal of Algebra, Combinatorics, Discrete Structures, and Applications, Sept. 2021; 8(3): 161-166
• Can a Subset's Topology Detect Continuous Extensions?  The College Mathematics Journal, 49(2): 138-139, March 2018.
• On cardinalities and compact closures.  Applied General Topology, 18(1): 107, April 2017, 107-115
• Kevin Cramer (CSULA student), Mike Krebs, Nicole Shabazi (CSULA student), Anthony Shaheen and Edward Voskanian (CSULA student). The Isoperimetric constant and Kazhdan constant of a Paley graph, Involve, 9(2): 293-306, March 2016.
• Al Fares, A. (CSULA student), E. Golvin (CSULA student), M. Krebs. A class of 2-groups of derived length three. Serdica Mathematical Journal, Volume 41, Numbers 2-3, 2015, pp. 329-332
• ZERO SUMS ON UNIT SQUARE VERTEX SETS, AND PLANE COLORINGS. Joint work with R. Katz and A. Shaheen. The American Mathematical Monthly. Vol. 121, No. 7 (August-September 2014), pp. 610-618
• Expander graphs. In J. Gross and J. Yellen (Eds.), Handbook of Graph Theory, Second Edition. CRC Press, 2013
• Expander families, group structure, and semidirect products. Joint work with M. Aivazian (CSULA student). Bulletin of the Irish Mathematical Society, Number 71 (2013), 21-30
• Simple examples of the combinatorial trace method in action. Joint work with N. Martinez (CSULA student). College Mathematics Journal, Vol. 44 (2013), no. 1, 32-36
• A combinatorial trace method: Counting closed paths to assay graph eigenvalues. Joint work with K. Dsouza (CSULA student). Rocky Mountain Journal of Mathematics, Vol. 43 (2013), no. 2, 469-478
• K-quasiderivations. (K-quasiderivations are maps that satisfy both the Product Rule and Chain Rule.) Central European Journal of Mathematics, Vol. 10 (2012), no. 2, 824-834
• Expander Families and Cayley Graphs: A Beginner's Guide, co-authored by A. Shaheen, Oxford University Press, New York, Oct. 2011
• mini-Sudokus and Groups. Joint work with C. Arcos (CSULA student) and G. Brookfield. Math. Mag. 83 (2010), no. 2, 111-122
• On Cantor's first uncountability proof, Pick's theorem, and the irrationality of the golden ratio. Joint work with T. Wright. Amer. Math. Monthly 117 (2010), no. 7, 633-637
• How to differentiate an integer modulo n. Joint work with C. Emmons and A. Shaheen. College Mathematics Journal, November 2009
• Faculty adviser for Fixed Points of Number Derivatives Modulo n by CSULA student F. Bains. Rose-Hulman Undergraduate Math Journal, Vol. 10 (2009), Issue 1
• Think Around The Box, a crossword puzzle of sorts. The Mathematical Intelligencer, Volume 31, Number 2, 59-60
• On the Spectra of Johnson Graphs. Joint work with A. Shaheen. (Appears in Electronic Journal of Linear Algebra, Volume 17, pp. 154-167, March 2008)
• Toledo invariants of Higgs bundles on elliptic surfaces associated to base orbifolds of Seifert fibered homology 3-spheres. (Appears in Michigan Math. J. Volume 56, Issue 1 (2008), 3-27.)
• My Ph.D. dissertation

Master's theses supervised

• Sufficient Conditions for an Abelian Discrete Fundamental Group of a Cayley Graph with |S| Less Than or Equal to Three by Trent Green
• Standardized Abelian Cayley Graphs and Their Chromatic Numbers In Four Dimensions: A Partial Evaluation Of Case Two by Katherine Ortiz
• Standardized Abelian Cayley Graphs and Their Chromatic Numbers In Four Dimensions: A Partial Evaluation Of Case Three by Mason Meeks
• Standardized Abelian Cayley Graphs and Their Chromatic Numbers In Four Dimensions by Timothy Harris
• Kazhdan Constants for Finite Cyclic Groups by Travis Hayes
• Kazhdan Constants Associated with Isomorphic Cayley Graphs by Manuel Davila
• To what Extent is the Kazhdan Constant a Graph Invariant? by Marcos Reyes
• Semidirect product of the integers modulo two by Ahmed Al-Fares
• Lower bounds on the orders of the terms of the derived series of finite 2-groups of derived length 3 by Ezekiel Golvin
• Utilizing wreath products to construct a sequence of Cayley Graphs with logarithmic diameter by Preston Smith
• Iterated semi-direct product of the integers modulo two by Matthew Aivazian
• Subspace methods in the eigenvalue problem by Yasmine Balkis
• The Wirtinger Presentation of the Fundamental Group of A Knot Complement by Chris Thiel
• A master's thesis on diameters of Cayley graphs by Keith D'Souza
• A master's thesis on Krull dimension by Carlos Arcos
• A master's thesis on Ramanujan graphs by Mustafa El-Sanfaz

Presentations

• How to Put Rainbow Sprinkles on a Donut: Chromatic Numbers of Abelian Cayley Graphs

Other, random stuff

• A group of Cal State LA students on the My Favorite Theorem podcast
• Silly math videos, including some that I've made
• Bizarre and/or mediocre songs I've made
• Trivia questions from my quizzes and tests

I graduated from Johns Hopkins University in May 2005. My thesis advisor was Dr. Richard Wentworth.

This website was last updated in May 2022.