In the classical works [1], [2] representations of quivers (in the category of finite-dimensional vector spaces) were considered and their connection with root systems of corresponding graphs was established, which developing further in the works [3]- [6]. Repeatedly attempts were made to generalize representations of quivers to metric and, in particular, Hilbert spaces, but at that mentioned connection was lost. In [7] a restriction of local scalarity was imposed on representations of graphs in Hilbert spaces, and after that it was managed to generalize the results of [1], [2] in a natural way by constructing, analogous to Coxeter functors, functors of even and add reflections, moreover it turned out that just these representations are of interest of functional analysis and in some particular cases in other terms in fact were considered in [8]- [9].In this paper we will consider a connection of locally-scalar representations of extended Dynkin graphs with function ρ considered in [10], and also functions ρ k playing analogous part for more wide class of graphs.

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We consider the connection of the separating functions ρ r with locally scalar representations of graphs and with spectral theory of graphs. Separating Functions ρ ρ ρ ρ rThe present paper is devoted to the investigation of the connection of the separating functions ρ r introduced in [1] with locally scalar representations of graphs, on the one hand, and with spectral theory of graphs, on the other.A function P ( S ) that associates a partially ordered set S with a positive rational number was introduced in [2]. In terms of the function P, a criterion for the finite representability and tameness of an arbitrary partially ordered set was formulated in [2]: a partially ordered set S is representable (tame) if and only if P ( S ) < 4 ( P ( S ) = 4). If a partially ordered set S is the union of s disjoint chains (i.e., chains such that any two elements belonging to different chains are incomparable), then P ( S ) = ρ ( n 1 , n 2 , … , n s ) = i s i n = ∑ 1 ρ( ) = i s = ∑ 1

We investigate the finite-dimensionality and growth of algebras specified by a system of polylinearly interrelated generators. The results obtained are formulated in terms of a function ρ.Assume that V = V m is an m-dimensional vector space over a certain field k with basis ( e 1 , … , e m ) ,A free associative algebra T m with m generators can be regarded as a tensor algebra T ( V m ). Let I = I ( ψ 1 , … , ψ s ) denote a (two-sided) ideal in T m generated by elements ψ 1 , … , ψ s .In the present paper, we study the problem of finite-dimensionality and growth of the algebras T m / I, depending on m, s, and the polynomials ψ i , under a certain restriction on the form of the polynomials ϕ i , which is, apparently, inessential.Assume that k = Q ( Σ ) is an infinite purely transcendental extension of a field Q obtained by the adjunction to Q of an algebraically independent countable set Σ, n 1 , … , n s ∈ N, n i ≤ n j for i < j, and ϕ i = x n i +1 , i = 1, s. Further, let M be a matrix whose rows are vectors v 1 , … , v s . It is easy to see that elementary transformations of columns of the matrix M as well as permutations of its rows and the multiplication of an arbitrary row by a nonzero element q ∈ k do not change (up to an isomorphism) the algebra T m / I ( ψ 1 , … , ψ s ).If s ≤ m, then the problem of finite-dimensionality and growth of the algebras T m / I = T m / I ( ψ 1 , … , ψ s ) is trivial:(a) the algebra T m / I has polynomial growth for s = m = 2 and n 1 = n 2 = 1, (b) in the other cases, T m / I has exponential growth.Let s > m. The matrix M can be reduced to the formwhere E is the m × m identity matrix and S is a certain r × m matrix ( r = s -m ) over k. Thus, we can set v i = e i for i = 1, m.

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