Abstract
Sturm-Liouville problem with nonlocal boundary conditions arises in many scientific fields such as chemistry, physics, or biology. There could be found some references to graph theory in a discrete Sturm-Liouville problem, especially in investigation of spectrum curves. In this paper, relations between discrete Sturm-Liouville problem with nonlocal boundary conditions characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found.
Highlights
For every Constant Eigenvalue Point (CEP) cj we define nonregular Spectrum Curve Nj = {cj}
Two or more Spectrum Curves may intersect at CP
CPs, regular and nonregular Spectrum Curves, CEPs were found by Bingelė [1]
Summary
Particular properties of the spectrum of a discrete Sturm–Liouville. Problem (dSLP) [1, 5, 6] with Nonlocal Boundary Conditions (NBCs) were found using Euler’s charakteristic formula [4]. Z h (q ) γc(q) = γc(q; ξ) := Pξh(q) , q ∈ Chq. All nonconstant eigenvalues (which depend on the parameter γ) are γ-points of (Complex-Real) Characteristic Function (CF)[7]. If q ∈ Dξ and γc(q) = 0 (q is not a Critical Point (CP) of CF), Eξ(γ) is smooth parametric curve N : R → Chq locally and we can add arrow on this curve (arrows show the direction in which γ ∈ R is increasing) We call such curves regular Spectrum Curves [2]. We denote a set Poles P := {pi, i = 1, np}, where np is the number of poles at Chq. For our problems P ⊂ Rhx ∪ {0} and all poles are of the first order (we write deg+(p) = 1, p ∈ P). The ordered pair is called weakly connected if an undirected path leads from v1 to v2 after replacing all of its directed arrows with undirected edges
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