A lower bound for the exponential Randić index of line graphs of trees

The exponential Randić index of a graph <span class="math inline">\(G\)</span>, denoted by <span class="math inline">\(ER(G)\)</span>, is defined as <span class="math inline">\(\sum\limits_{uv\in E(G)}e^{\frac{1}{d(u)d(v)}}\)</span>, where <span class="math inline">\(d(u)\)</span> denotes the degree of a vertex <span class="math inline">\(u\)</span> in <span class="math inline">\(G\)</span>. The line graph <span class="math inline">\(L(G)\)</span> of a graph <span class="math inline">\(G\)</span> is a graph in which each vertex represents an edge of <span class="math inline">\(G\)</span>, and two vertices are adjacent in <span class="math inline">\(L(G)\)</span> if and only if their corresponding edges in <span class="math inline">\(G\)</span> are incident to a common vertex. In this paper, we proved that for any tree <span class="math inline">\(T\)</span> of order <span class="math inline">\(n\ge3\)</span>, <span class="math inline">\(ER(L(T))>\frac{n}{4}e^{\frac{1}{2}}\)</span>.