| Summary: | <pre><span>Let </span><span class="math inline"><em>G</em></span><span> be a finite group. For a fixed element </span><span class="math inline"><em>g</em></span><span> in </span><span class="math inline"><em>G</em></span><span> and a given subgroup </span><span class="math inline"><em>H</em></span><span> of </span><span class="math inline"><em>G</em></span><span>, the relative </span><span class="math inline"><em>g</em></span><span>-noncommuting graph of </span><span class="math inline"><em>G</em></span><span> is a simple undirected graph whose vertex set is </span><span class="math inline"><em>G</em></span><span> and two vertices </span><span class="math inline"><em>x</em></span><span> and </span><span class="math inline"><em>y</em></span><span> are adjacent if </span><span class="math inline"><em>x</em> ∈ <em>H</em></span><span> or </span><span class="math inline"><em>y</em> ∈ <em>H</em></span><span> and </span><span class="math inline">[<em>x</em>, <em>y</em>]≠<em>g</em>, <em>g</em><sup>−1</sup></span><span>. We denote this graph by </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span>. In this paper, we obtain computing formulae for degree of any vertex in </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span> and characterize whether </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span> is a tree, star graph, lollipop or a complete graph together with some properties of </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span> involving isomorphism of graphs. We also present certain relations between the number of edges in </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span> and certain generalized commuting probabilities of </span><span class="math inline"><em>G</em></span><span> which give some computing formulae for the number of edges in </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span>. Finally, we conclude this paper by deriving some bounds for the number of edges in </span><span class="math inline"><em>Γ</em><sub><em>H</em>, <em>G</em></sub><sup><em>g</em></sup></span><span>.</span></pre>
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