The Total Number of Matchings in Triangle Graph of a Connected Graph

Abstract

For a graph \(G\), let \({Z}(G)\) be the total number of matchings in \(G\). For a nontrivial graph \(G\) of order \(n\) with vertex set \(V(G) = \{v_1, \ldots, v_n\}\), Cvetković et al. \([2]\) defined the triangle graph of \(G\), denoted by \(R(G)\), to be the graph obtained by introducing a new vertex \(v_{ij}\) and connecting \(u_{ij}\) both to \(v_i\) and to \(v_j\) for each edge \(v_iv_j\) in \(G\). In this short paper, we prove that for a connected graph \(G\), if \({Z}(R(G)) \geq (\frac{1}{2}n-\frac{1}{2}+\frac{5}{2n})^2\), then \(G\) is traceable. Moreover, for a connected graph \(G\), we give sharp upper bounds for \({Z}(R(G))\) in terms of clique number, vertex connectivity, and spectral radius of \(G\), respectively.