Publications

Properties of a q-analogue for zero forcing

Published in Graphs Combin., 2020

This paper investigates a q-analogue of zero forcing. Basic properties of this game are established including determining all graphs which have minimal cost 1 or 2 for all possible q, and finding the zero forcing number for all trees when q=1.

Recommended citation: S. Butler, C. Erickson, S. Fallat, H.T. Hall, B. Kroschel, J.C.H. Lin, B. Shader, N. Warnberg, B. Yang. “Properties of a q-analogue for zero forcing.” Graphs Combin. 36 (2020), No. 5, 1401—1419. https://arxiv.org/abs/1809.07640

Restricted power domination and zero forcing problems

Published in J. Comb. Optim., 2019

A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X.

Recommended citation: C. Bozeman, B. Brimkov, C. Erickson, D. Ferrero, M. Flagg, L. Hogben. "Restricted power domination and zero forcing problems." J. Comb. Optim. 37 (2019), No. 3, 935—956. https://arxiv.org/abs/1711.05190

Rainbow arithmetic progressions

Published in J. Comb., 2016

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n], k) denotes the smallest number of colors with which the integers {1,…,n} can be colored and still guarantee there is a rainbow arithmetic progression of length k.

Recommended citation: S. Butler, C. Erickson, L. Hogben, K. Hogenson, L. Kramer, R.L. Kramer, J.C.-H. Lin, R.R. Martin, D. Stolee, N. Warnberg, M. Young. "Rainbow arithmetic progressions." J. Comb. 7 (2016), No. 4, 595—626. https://doi.org/10.4310/JOC.2016.v7.n4.a3

Sign patterns that require eventual exponential nonnegativity

Published in Electron. J. Linear Algebra, 2015

Sign patterns that require exponential nonnegativity are characterized. A set of conditions necessary for a sign pattern to require eventual exponential nonnegativity are established. It is shown that these conditions are also sufficient for an upper triangular sign pattern to require eventual exponential nonnegativity and it is conjectured that these conditions are both necessary and sufficient for any sign pattern to require eventual exponential nonnegativity. It is also shown that the maximum number of negative entries in a sign pattern that requires eventual exponential nonnegativity is (n-1)(n-2)/2 + 2.

Recommended citation: C. Erickson. "Sign patterns that require eventual exponential nonnegativity." Electron. J. Linear Algebra 30 (2015), 171--195. https://doi.org/10.13001/1081-3810.3027

Positive semidefinite zero forcing

Published in Linear Algebra Appl., 2013

We establish a variety of properties of positive semidefinite zero forcing: Any vertex of G can be in a minimum positive semidefinite zero forcing set (this is not true for standard zero forcing). Graphs having extreme values of the positive semidefinite zero forcing number are characterized. The effect of various graph operations on positive semidefinite zero forcing number and connections with other graph parameters are studied.

Recommended citation: J. Ekstrand, C. Erickson, H.T. Hall, D. Hay, L. Hogben, R. Johnson, N. Kingsley, S. Osborne, T. Peters, J. Roat, A. Ross, D.D. Row, N. Warnberg, and M. Young. “Positive semidefinite zero forcing.” Linear Algebra Appl. 439 (2013), 1862–1874. https://doi.org/10.1016/j.laa.2013.05.020

Potentially eventually exponentially positive sign patterns

Published in Involve, 2013

We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP sign patterns. A characterization of all 2-by-2 and 3-by-3 PEEP sign patterns is given.

Recommended citation: M. Archer, M. Catral, C. Erickson, R. Haber, L. Hogben, X. Martinez-Rivera, and A. Ochoa. "Potentially eventually exponentially positive sign patterns." Involve 6 (2013), No. 3, 261—271. https://doi.org/10.2140/involve.2013.6.261

Sign patterns that allow strong eventual nonnegativity

Published in Electron. J. Linear Algebra, 2012

A new class of sign patterns contained in the class of sign patterns that allow eventual nonnegativity is introduced and studied. A sign pattern is potentially strongly eventually nonnegative (PSEN) if there is a matrix with this sign pattern that is eventually nonnegative and has some power that is both nonnegative and irreducible. Using Perron-Frobenius theory and a matrix perturbation result, it is proved that a PSEN sign pattern is either potentially eventually positive or r-cyclic. The minimum number of positive entries in an n-by-n PSEN sign pattern is shown to be n, and PSEN sign patterns of orders 2 and 3 are characterized.

Recommended citation: M. Catral, C. Erickson, L. Hogben, D.D. Olesky, and P. Van den Driessche. "Sign patterns that allow strong eventual nonnegativity." Electron. J. Linear Algebra 23 (2012), 1—10. https://doi.org/10.13001/1081-3810.1502

Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees

Published in Electron. J. Linear Algebra, 2012

The definition of the positive semidefinite zero forcing number for simple graphs is extended to multigraphs; as for simple graphs, this parameter bounds the maximum positive semidefinite nullity from above.

Recommended citation: J. Ekstrand, C. Erickson, D. Hay, L. Hogben, and J. Roat. "Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees." Electron. J. Linear Algebra 23 (2012), 79—87. https://doi.org/10.13001/1081-3810.1506

Constructions of potentially eventually positive sign patterns with reducible positive part

Published in nvolve, 2011

Potentially eventually positive (PEP) sign patterns were introduced by Berman et al. (Electron. J. Linear Algebra 19 (2010), 108–120), where it was noted that a matrix is PEP if its positive part is primitive, and an example was given of a 3-by-3 PEP sign pattern with reducible positive part. We extend these results by constructing n-by-n PEP sign patterns with reducible positive part, for every n ≥ 3 .

Recommended citation: M. Archer, M. Catral, C. Erickson, R. Haber, L. Hogben, X. Martinez-Rivera, and A. Ochoa. "Constructions of potentially eventually positive sign patterns with reducible positive part." Involve 4 (2011), No. 4, 405—410. https://doi.org/10.2140/involve.2011.4.405

On nilpotence indices of sign patterns

Published in Commun. Korean Math. Soc., 2010

The work in this paper was motivated by Eschenbach and Li, who listed four 4-by-4 sign patterns, conjectured to be nilpotent sign patterns of nilpotence index at least 3. These sign patterns with no zero entries, called full sign patterns, are shown to be potentially nilpotent of nilpotence index 3. We also generalize these sign patterns of order 4 so that we provide classes of n-by-n sign patterns of nilpotence indices at least 3, if they are potentially nilpotent. Furthermore it is shown that if a full sign pattern A of order n has nilpotence index k with 2≤k≤n−1, then sign pattern A has nilpotent realizations of nilpotence indices k, k+1,…,n.

Recommended citation: C. Erickson and I.-J. Kim. ”On nilpotence indices of sign patterns." Commun. Korean Math. Soc. 25 (2010), No. 1, 11—18. https://doi.org/10.4134/CKMS.2010.25.1.11

Craig Erickson

Publications