Abstract：Circulant graphs are an important class of interconnection networks in parallel and distributed computing, and integral circulant graphs play an important role in modeling quantum spin networks which support the perfect state transfer. The rank of a graph is defined to be the rank of its adjacency matrix. In this note, using Ramanujan sums, by means of the Euler function and the Mobius function, we study the rank for some integral circulant graphs, and deduce the bounds of rank for integral circulant graphs.
周后卿. 几类整循环图的秩的界[J]. 浙江大学学报（理学版）, 2020, 47(3): 301-305.
ZHOU Houqing. Bounds of rank for some integral circulant graphs. Journal of ZheJIang University(Science Edition), 2020, 47(3): 301-305.
1 CHANG G J, HUANG L H, YEH H G. On the rank of a cograph[J]. Linear Algebra and Its Applications, 2008, 429(2/3): 601-605. 2 CHANG G J, HUANG L H, YEH H G. A characterization of graphs with rank 4[J]. Linear Algebra and Its Applications, 2011,434:1793-1798. 3 CHANG G J, HUANG L H, YEH H G. A characterization of graphs with rank 5[J]. Linear Algebra and Its Applications, 2012,436(11): 4241-4250. 4 CHEN L. Revised version on possible rank of reduced unicyclic graphs with a given order[J]. Mathematica Applicata, 2014,27(1): 214-220. 5 苏莉,李红海,陈春芳.具有较小秩的带号图的刻画[J].浙江大学学报(理学版),2013,40(6): 611-617,626. DOI:10.3785/j.issn.1008-9497.2013.06.01 SU L, LI H H, CHEN C F. A characterization of signed graphs with small rank[J]. Journal of Zhejiang University(Science Edition), 2013,40(6): 611-617,626. DOI:10.3785/j.issn.1008-9497.2013.06.01 6 周后卿. 网络拓扑的超能整循环图构造[J]. 计算机工程与应用,2016,52(9): 23-27,32. ZHOU H Q. Construction of hyperenergetic integral circulant graphs for network topology[J]. Computer Engineering and Applications, 2016, 52(9):23-27, 32. 7 DAVIS P J. Circulant Matrices[M]. New York: John Wiley & Sons,1979. 8 WASIN S. Integral circulant graphs[J]. Discrete Mathematics, 2006, 306(1):153-158. 9 MOLLAHAJIAGHAEI M. The eigenvalues and energy of integral circulant graphs[J]. Transactions on Combinatorics, 2012,1(3): 47-56.
2020, Vol. 47 
