Abstract. Polynomials on stranded graphs are higher dimensional generalization of Tutte and Bollobás-Riordan polynomials [Math. Ann. 323 (2002), 81-96]. Here, we deepen the analysis of the polynomial invariant defined on rank 3 weakly-colored stranded graphs introduced in arXiv:1301.1987. We successfully find in dimension D ≥ 3 a modified Euler characteristic with D − 2 parameters. Using this modified invariant, we extend the rank 3 weakly-colored graph polynomial, and its main properties, on rank 4 and then on arbitrary rank D weakly-colored stranded graphs.

In this paper, we analyze the Bollobas and Riordan polynomial for ribbon graphs with flags introduced in arXiv:1301.1987[math.CO] and prove its universality. We also show that this polynomial can be defined on some equivalence classes of ribbon graphs involving flag moves and that the new polynomial is still universal on these classes.

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This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic freedom of the model. We then define the Connes-Kreimer-like Hopf algebra describing the combinatorics of the renormalization of this model and we analyze in detail, at one-and two-loop levels, the Hochschild cohomology allowing to write the combinatorial Dyson-Schwinger equations. Feynman tensor graph Hopf subalgebras are also exhibited.