Open problem on the Euler Sombor index of trees with given domination number

<p>The Euler Sombor (<span class="math inline">\(EU\)</span>) index of a graph <span class="math inline">\(G\)</span> is defined as <span class="math display">\[EU(\mathit{G})=\sum \limits_{{\mathit{u}}{\mathit{v}}\in E(\mathit{G})} \sqrt{deg_G^2(u)+deg_G^2(v)+deg_G(u)deg_G(v)},\]</span> where <span class="math inline">\(deg_G(u)\)</span> and <span class="math inline">\(deg_G(v)\)</span> are the degrees of the vertices <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span> in the graph <span class="math inline">\(G\)</span>, respectively. Biswaranja Khanra and Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem., 94 (2025) 525–548] posed an open problem to determine the extremal values and extremal graphs of the Euler Sombor index in the class of all connected graphs with a given domination number. In this paper, we solve this open problem for trees with a given domination number. Furthermore, we determine an upper bound for the Euler Sombor index of trees with a given independence number. We also characterize the corresponding extremal trees. Additionally, we propose a set of open problems for future research.</p>