Results on the elliptic Sombor index

The newly introduced elliptic Sombor index of a graph G is defined as follows <ns0:disp-formula id="FD1"><ns0:alternatives><ns0:textual-form specific-use="web-only">ESO(G) = ∑uvϵE (d(u) + d(v))√d(u)2 + d(v)2</ns0:textual-form><ns5:math display="block" id="mml_eq1"><ns5:mi>ESO</ns5:mi><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>G</ns5:mi><ns5:mo stretchy="false">)</ns5:mo><ns5:mo>=</ns5:mo><ns5:munder><ns5:mo>∑</ns5:mo><ns5:mrow><ns5:mi>uvϵ</ns5:mi><ns5:mrow><ns5:mi>E</ns5:mi></ns5:mrow></ns5:mrow></ns5:munder><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>d</ns5:mi><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>u</ns5:mi><ns5:mo stretchy="false">)</ns5:mo><ns5:mo>+</ns5:mo><ns5:mi>d</ns5:mi><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>v</ns5:mi><ns5:mo stretchy="false">)</ns5:mo><ns5:mo stretchy="false">)</ns5:mo><ns5:msqrt><ns5:mi>d</ns5:mi><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>u</ns5:mi><ns5:msup><ns5:mo stretchy="false">)</ns5:mo><ns5:mn>2</ns5:mn></ns5:msup></ns5:msqrt><ns5:mo>+</ns5:mo><ns5:mrow><ns5:mi>d</ns5:mi><ns5:mo stretchy="false">(</ns5:mo><ns5:mi>v</ns5:mi><ns5:msup><ns5:mo stretchy="false">)</ns5:mo><ns5:mn>2</ns5:mn></ns5:msup></ns5:mrow><ns5:mo>,</ns5:mo></ns5:math><ns0:tex-math id="tex_eq1">$$ ESO (G) = \sum_{u v \epsilon {E}} (d(u) + d(v))\sqrt{d(u)^2} + {d(v)^2}, $$</ns0:tex-math><ns0:graphic id="img_eq1" mime-subtype="gif" mimetype="image" ns4:href="ro240291-eq1.gif"></ns0:graphic></ns0:alternatives></ns0:disp-formula> in which E and d(u) are the edge set of G and the degree of the vertex u in G, respectively. In this paper, we compute the elliptic Sombor index for certain graphs. Furthermore, we obtain new results and bounds for the elliptic Sombor index in a graph.