| Summary: | <p>Given a group <span class="math"><em>G</em></span>, the intersection power graph of <span class="math"><em>G</em></span>, denoted by <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span>, is the graph with vertex set <span class="math"><em>G</em></span> and two distinct vertices <span class="math"><em>x</em></span> and <span class="math"><em>y</em></span> are adjacent in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if there exists a non-identity element <span class="math"><em>z</em> ∈ <em>G</em></span> such that x<sup>m</sup>=z=y<sup>n</sup>, for some <span class="math"><em>m</em>, <em>n</em> ∈ N</span>, i.e. <span class="math"><em>x</em> ∼ <em>y</em></span> in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if <span class="math">⟨<em>x</em>⟩ ∩ ⟨<em>y</em>⟩ ≠ {<em>e</em>}</span> and <span class="math"><em>e</em></span> is adjacent to all other vertices, where <span class="math"><em>e</em></span> is the identity element of the group <span class="math"><em>G</em></span>. Here we show that the graph <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is complete if and only if either <span class="math"><em>G</em></span> is cyclic <span class="math"><em>p</em></span>-group or <span class="math"><em>G</em></span> is a generalized quaternion group. Furthermore, <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is Eulerian if and only if <span class="math">∣<em>G</em>∣</span> is odd. We characterize all abelian groups and also all non-abelian <span class="math"><em>p</em></span>-groups <span class="math"><em>G</em></span>, for which <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is dominatable. Beside, we determine the automorphism group of the graph <span class="math">G<sub><em>I</em></sub>(Z<sub><em>n</em></sub>)</span>, when <span class="math"><em>n</em> ≠ <em>p</em><sup><em>m</em></sup></span>.</p>
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