• **Entropy related to** *IRM* **index**

Let *<sup>μ</sup>*((*ai*)(*aj*)) = [*Vai* − *Vaj* ] 2 . Then the *IRM* entropy (4) is given by

$$IRM\_{(G)} \quad = \sum\_{a\_i, a\_j \in \mathbb{Z}\_G} \left\{ \left[ V\_{a\_i} - V\_{a\_j} \right]^2 \right\} = \sum\_{a\_i, a\_j \in \mathbb{Z}\_G} \mu(\left( a\_i \right)(a\_j)).$$

Adding the parameters of *IRM*(*G*) into Equation (5), then the IRM (*ENTIRM*) entropy is

$$ENT\_{IRM\_{(G,x)}} = \log IRM\_{(G)} - \frac{1}{IRM\_{(G,x)}} \log \left\{ \prod\_{a\_i, a\_j \in \tilde{\xi}\_G} \left( [V\_{a\_i} - V\_{a\_j}]^2 \right)^{\left( [V\_{a\_i} - V\_{a\_j}]^2 \right)} \right\}.\tag{10}$$
