Euler-Sombor index and its congeners

<p>The Euler-Sombor index <span class="math inline">\(EU\)</span> is a vertex-degree-based graph invariant, defined as the sum over all pairs of adjacent vertices <span class="math inline">\(u,v\)</span> of the underlying graph, of the terms <span class="math inline">\(\sqrt{d_u^2+d_v^2+d_u\,d_v}\)</span>, where <span class="math inline">\(d_u\)</span> and <span class="math inline">\(d_v\)</span> are the degrees of the vertices <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span>, respectively. For a real number <span class="math inline">\(\lambda\)</span>, a variable version of <span class="math inline">\(EU\)</span> is constructed, denoted by <span class="math inline">\(EU(\lambda)\)</span>, defined via <span class="math inline">\(\sqrt{d_u^2+d_v^2+\lambda\,d_u\,d_v}\)</span>. Its special cases for <span class="math inline">\(\lambda=2,\,-2,\,0\)</span>, and 1 are, respectively, the first Zagreb, Albertson, Sombor, and the ordinary Euler-Sombor indices. The basic properties of <span class="math inline">\(EU(\lambda)\)</span> are determined, including a method for its approximate calculation and bounds in terms of minimum degree, maximum degree, order and size for several graph products. It is shown how to find values of <span class="math inline">\(\lambda\)</span> for which <span class="math inline">\(EU(\lambda)\)</span> is optimal with regard to predicting molecular properties.</p>