Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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MINIMUM RANK OF THE LINE GRAPH OF CORONA CnoKt

  • Im, Bokhee (Department of Mathematics Chonnam National University) ;
  • Lee, Hwa-Young (Department of Mathematics Chonnam National University)
  • Received : 2015.01.14
  • Published : 2015.04.30
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Abstract

The minimum rank mr(G) of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose (i, j)-th entry (for $i{\neq}j$) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The corona $C_n{\circ}K_t$ is obtained by joining all the vertices of the complete graph $K_t$ to each n vertex of the cycle $C_n$. For any t, we obtain an upper bound of zero forcing number of $L(C_n{\circ}K_t)$, the line graph of $C_n{\circ}K_t$, and get some bounds of $mr(L(C_n{\circ}K_t))$. Specially for t = 1, 2, we have calculated $mr(L(C_n{\circ}K_t))$ by the cut-vertex reduction method.

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References

  1. AIM Minimum Rank-Special Graphs Work Group(1-AIM), Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008), no. 7, 1628-1648. https://doi.org/10.1016/j.laa.2007.10.009
  2. E. Almodovar, L. DeLoss, L. Hogben, K. Hogenson, K. Murphy, T. Peters, and C. A. Ramrez, Minimum rank, maximum nullity and zero forcing number for selected graph families, Involve 3 (2010), no. 4, 371-392. https://doi.org/10.2140/involve.2010.3.371
  3. F. Barioli, S. M. Fallat, and L. Hogben, Computation of minimal rank and path cover number for certain graphs, Linear Algebra Appl. 392 (2004), 289-303. https://doi.org/10.1016/j.laa.2004.06.019
  4. S. M. Fallat and L. Hogben, The minimum rank of symmetric matrices described by a graph: a survey, Linear Algebra Appl. 426 (2007), no. 2-3, 558-582. https://doi.org/10.1016/j.laa.2007.05.036
  5. L. Y. Hsieh, On minimum rank matrices having prescribed graph, Ph.D. Thesis, University of Wisconsin-Madison, 2001.
  6. P. M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996), 303-316. https://doi.org/10.1016/0024-3795(95)00238-3