Triply Equienergetic Graphs

Pairs of non-cospectral graphs are constructed, having equal energy (E), Laplacian energy (LE), and distance energy (DE). These seem to be the first examples of “triply equienergetic graphs”. We construct a family of integral circulant graphs (ICG )o f order n =2 pq , where p>q >2 are prime numbers, Gn = ICG(n, {1, 2}) and Hn = ICG(n, {p, 2p, q, 2q}) , for which E(Gn )= E(Hn )=8 (p − 1)(q − 1) , LE(Gn )= LE(Hn )= 8(p − 1)(q − 1) , and DE(Gn )= DE(Hn )=8 (p − 1)(q − 1) + 4pq .

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