Mantel’s theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order n. In 1970, Nosal obtained a spectral version of Mantel’s theorem which gave the maximum spectral radius of a triangle-free graph of order n. In this paper, the clique tensor of a graph G is proposed and the spectral Mantel’s theorem is extended via the clique tensor. Furthermore, a sharp upper bound of the number of cliques in G via the spectral radius of the clique tensor is obtained. These results imply that a result of Erdős (Magyar Tud Akad Mat Kutató Int Közl 7:459–464, 1962) under certain conditions.

曼特尔定理是极值图论中的经典结果，它意味着n阶无三角形图的最大边数。1970 年，诺萨尔获得了曼特尔定理的谱版本，该定理给出了n阶无三角形图的最大谱半径。本文提出了图G的团张量，并通过团张量推广了谱曼特尔定理。此外，通过团张量的谱半径获得了G中团数量的尖锐上限。这些结果意味着 Erdős (Magyar Tud Akad Mat Kutató Int Közl 7:459–464, 1962) 在某些条件下的结果。