Summary: The energy \(\mathcal{E}(G)\) of a graph \(G\) is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. A graph \(G\) of order \(n\) is said to be borderenergetic if its energy equals the energy of the complete graph \(K_n\), i.e., if \(\mathcal{E}(G) = 2(n-1)\). We first show by examples that there exist connected borderenergetic graphs, different from the complete graph \(K_n\). The smallest such graph is of order 7. We then show that for each integer \(n\), \(n\ge 7\), there exists borderenergetic graphs of order \(n\), different from \(K_n\), and describe the construction of some of these graphs.