Minimum rank of the line graph of corona $C_n\circ K_t$
Commun. Korean Math. Soc. 2015 Vol. 30, No. 2, 65-72
https://doi.org/10.4134/CKMS.2015.30.2.65
Printed April 30, 2015
Bokhee Im and Hwa-Young Lee
Chonnam National University, Chonnam National University
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.
Keywords : minimum rank, zero forcing, line graph, corona, ciclo
MSC numbers : Primary 05C50; Secondary 15A03
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