Metrically regular bigraphs the square of which are metrically regular graphs are investigated in the case of graphs with 6 distinct eigenvalues (these eigenvalues can have variuos multiplicities).
The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only one table of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter (7 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter see [7] and [8].
We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.
Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied...
We provide a construction of monomial ideals in such that , where denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on that generalize some results...
In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of . Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.
The Hopf algebra of word-quasi-symmetric functions (), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on . This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret as a convolution algebra of linear endomorphisms of quasi-shuffle...