| Summary: | <p>The distance signless Laplacian spectral radius of a connected graph <span class="math"><em>G</em></span> is the largest eigenvalue of the distance signless Laplacian matrix of <span class="math"><em>G</em></span>, defined as <span class="math"><em>D</em><sup><em>Q</em></sup>(<em>G</em>) = <em>T</em><em>r</em>(<em>G</em>) + <em>D</em>(<em>G</em>)</span>, where <span class="math"><em>D</em>(<em>G</em>)</span> is the distance matrix of <span class="math"><em>G</em></span> and <span class="math"><em>T</em><em>r</em>(<em>G</em>)</span> is the diagonal matrix of vertex transmissions of <span class="math"><em>G</em></span>. In this paper we determine some upper and lower bounds on the distance signless Laplacian spectral radius of <span class="math"><em>G</em></span> based on its order and independence number, and characterize the extremal graph. In addition, we give an exact description of the distance signless Laplacian spectrum and the distance signless Laplacian energy of the join of regular graphs in terms of their adjacency spectrum.</p>
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