A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex outside $D$ has a neighbor in $D$. The domination number of $G$, denoted by $\gamma(G)$, is the minimum cardinality amongst all dominating sets of $G$. The domination entropy of $G$, denoted by $I_{dom}(G)$ is defined as $I_{dom}(G)=-\sum_{i=1}^k\frac{d_i(G)}{\gamma_S(G)}\log (\frac{d_i(G)}{\gamma_S(G)})$, where $\gamma_S(G)$ is the number of all dominating sets of $G$ and $d_i(G)$ is the number of dominating sets of cardinality $i$. A graph $G$ is $C_4$-free if it does not contain a $4$-cycle as a subgraph. In this note we first determine the domination entropy in the graphs whose complements are $C_4$-free. We then propose an algorithm that computes the domination entropy in any given graph. We also consider circulant graphs $G$ and determine $d_i(G)$ under certain conditions on $i$.
Ghameshlou, A. N. , Mohammadi, M. and JafariRad, A. (2025). A note on the domination entropy of graphs. Journal of Discrete Mathematics and Its Applications, 10(1), 11-20. doi: 10.22061/jdma.2025.11448.1106
MLA
Ghameshlou, A. N., , Mohammadi, M. , and JafariRad, A. . "A note on the domination entropy of graphs", Journal of Discrete Mathematics and Its Applications, 10, 1, 2025, 11-20. doi: 10.22061/jdma.2025.11448.1106
HARVARD
Ghameshlou, A. N., Mohammadi, M., JafariRad, A. (2025). 'A note on the domination entropy of graphs', Journal of Discrete Mathematics and Its Applications, 10(1), pp. 11-20. doi: 10.22061/jdma.2025.11448.1106
CHICAGO
A. N. Ghameshlou , M. Mohammadi and A. JafariRad, "A note on the domination entropy of graphs," Journal of Discrete Mathematics and Its Applications, 10 1 (2025): 11-20, doi: 10.22061/jdma.2025.11448.1106
VANCOUVER
Ghameshlou, A. N., Mohammadi, M., JafariRad, A. A note on the domination entropy of graphs. Journal of Discrete Mathematics and Its Applications, 2025; 10(1): 11-20. doi: 10.22061/jdma.2025.11448.1106
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