On path eigenvalues of some graphs

<p>Let <span class="math inline">\(G\)</span> be a graph with vertex set <span class="math inline">\(V(G) = \{v_1, v_2, \dots, v_n\}\)</span>. We associate to <span class="math inline">\(G\)</span>, a matrix <span class="math inline">\(P(G)\)</span> whose <span class="math inline">\((i, j)\)</span>-th entry is the maximum number of vertex-disjoint paths between the corresponding vertices if <span class="math inline">\(i\neq j\)</span>, and is zero otherwise. We call this matrix the <em>path matrix</em> of <span class="math inline">\(G\)</span>, and its eigenvalues are referred to as the <em>path eigenvalues</em> of <span class="math inline">\(G\)</span>. In this paper, we investigate the path eigenvalues of graphs resulting from certain graph operations and specific graph families.</p>