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In this paper, we consider Ising-Vannimenus model on a Cayley tree for ordertwo with competing nearest-neighbor, prolonged next-nearest neighborinteractions. We stress that the mentioned model was investigated onlynumerically, without rigorous (mathematical) proofs. One of the main point ofthis paper is to propose a measure-theoretical approach the considered model.We find certain conditions for the existence of Gibbs measures corresponding tothe model. Then we establish the existence of the phase transition. Moreover,the free energies of the found Gibbs measures are calculated.

Gibbs measures and free energies of the Ising-Vannimenus Model on the Cayley tree