A generalization of a Turán’s theorem about maximum clique on graphs

Douglas Frederico Guimarães Santiago (Institute of Science and Technology (ICT), Federal University of the Jequinhonha and Mucuri Valleys, Diamantina, Brazil)
Anderson Luiz Pedrosa Porto (Institute of Science and Technology (ICT), Federal University of the Jequinhonha and Mucuri Valleys, Diamantina, Brazil)
Kaio Ariel Silva Sá (Institute of Science and Technology (ICT), Federal University of the Jequinhonha and Mucuri Valleys, Diamantina, Brazil)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications

One of the most important Turán’s theorems establishes an inequality between the maximum clique and the number of edges of a graph. Since 1941, this result has received much attention and many of the different proofs involve induction and a probability distribution. In this paper we detail finite procedures that gives a proof for the Turán’s Theorem. Among other things, we give a generalization of this result. Also we apply this results to a Nikiforov’s inequality between the spectral radius and the maximum clique of a graph.