Malaysian Journal of Mathematical Sciences, March 2025, Vol. 19, No. 1
On Graph Entropy Measures Based on the Number of Dominating and Power Dominating Sets
Geethu Kuriachan and Parthiban, A.
Corresponding Email: [email protected]
Received date: 14 June 2024
Accepted date: 22 November 2024
Abstract:
This article examines graph entropy measures that depend on the number of dominating and power-dominating sets. To quantify the structural complexity of a graph structure, one uses graph entropies. It is easy to compute these properties for smaller networks, and if reliable approximations are developed, similar metrics can also be used for larger graphs. Using various graph invariants, many graph entropy measures have already been established and computed. So, in this work, a new graph entropy measure, namely, power domination entropy, using the power domination polynomial, is introduced. The domination and power domination polynomials of graphs are used to determine the number of dominating and power dominating sets. Let $D(G, \xi)$ represent the collection of all dominating sets of $G$ with size $\xi$, $d_{\xi}(G) =\lvert D(G, \xi) \rvert$, and $\gamma_s$ be the total number of dominating sets of $G$. Then, the domination entropy of $G$ with $n$ nodes is defined as $I_{dom}(G) = -\sum\limits_{\xi=1}^{n}\dfrac{d_{\xi}(G)}{\gamma_s(G)} \ \log\left(\dfrac{d_{\xi}(G)}{\gamma_s(G)}\right)$. The domination and power domination entropies for a few graphs are further computed. Following that, a comparison between the domination and power domination entropies of several graphs is provided.
Keywords: graph entropy measures; domination; domination polynomial; power domination; power domination polynomial
