{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T13:54:11Z","timestamp":1648907651889},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2016,9]]},"abstract":"<jats:p> In this paper, we propose a new family of graphs, matrix graphs, whose vertex set [Formula: see text] is the set of all [Formula: see text] matrices over a finite field [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text]. And any two matrices share an edge if the rank of their difference is [Formula: see text]. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let [Formula: see text] (resp., [Formula: see text]) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most [Formula: see text] (resp., exactly [Formula: see text]) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of [Formula: see text] and give some upper and lower bounds on [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s1793830916500531","type":"journal-article","created":{"date-parts":[[2016,6,30]],"date-time":"2016-06-30T09:07:48Z","timestamp":1467277668000},"page":"1650053","source":"Crossref","is-referenced-by-count":0,"title":["Two coloring problems on matrix graphs"],"prefix":"10.1142","volume":"08","author":[{"given":"Zhe","family":"Han","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, University of Washington, Seattle, WA 98195, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mei","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. 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