{"title":"Minimal Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 3","authors":"Ber-Lin Yu, Ting-Zhu Huang","country":null,"institution":"","volume":37,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":127,"pagesEnd":130,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/11586","abstract":"

If there exists a nonempty, proper subset S of the set of all (n + 1)(n + 2)\/2 inertias such that S Ôèå i(A) is sufficient for any n × n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In , Kim, Olesky and Driessche identified all minimal critical sets of inertias for 2 × 2 zero-nonzero patterns. Identifying all minimal critical sets of inertias for n × n zero-nonzero patterns with n ≥ 3 is posed as an open question in . In this paper, all minimal critical sets of inertias for 3 × 3 zero-nonzero patterns are identified. It is shown that the sets {(0, 0, 3), (3, 0, 0)}, {(0, 0, 3), (0, 3, 0)}, {(0, 0, 3), (0, 1, 2)}, {(0, 0, 3), (1, 0, 2)}, {(0, 0, 3), (2, 0, 1)} and {(0, 0, 3), (0, 2, 1)} are the only minimal critical sets of inertias for 3 × 3 irreducible zerononzero patterns.<\/p>\r\n","references":null,"publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 37, 2010"}