arXiv subjects
Publications and source records attributed to I. K. Redchuk.
In the present paper we show that the spectrum of an arbitrary starlike graph can be completely determined via separating functions $ρ_t$ (see \cite{NazRoi,RedRoi,Red3}). This fact helps to get in an easy way several results for the spectra of starlike graphs.
In this article we consider the connection of separating finctions $ρ_r$ with locally scalar representations of graphs and with spectral graph theory.
A numeric function $ρ$: $ρ(k)=1+\frac{k-1}{k+1}, k \in N$ was considered in [1]. In its terms criterions of finite representability and tameness of marked quivers, posets with equivalence and dyadic posets can be obtained; Dynkin schemes and extended schemes also can be characterized. In this paper authors consider the connection of function $ρ$ with locally-scalar representations [2] of extended Dynkin graphs. Then a family of functions $ρ_n$ is defined -- a generalization of function $ρ$, which plays an analogous part for more wide class of graphs. Also some properties of functions $ρ$ and $ρ_k$ are proved. References [1] L.A. Nazarova, A.V. Roiter. {\it Norm of a relation, separating functions and representations of marked quivers.} Ukr. Math. Jour., 54(2002), No.6, p.808-840. [2] S.A. Kruglyak, A.V. Roiter. {\it Locally-scalar representations of graphs in the category of Hilbert spaces.} Prepr. Ukr. Math. Jour. (2003).