1 A discrete Sturm–Liouville Problem

In this paper, 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].

We introduce a uniform grid and we use notation ${\bar{w}}^h=\{t_j=jh,j=0,...,n;nh=1\}$ for $2<n∊\mathbb{N},$ and ${\mathbb{N}}^h:=(0,n)\cap \mathbb{N},{\bar{\mathbb{N}}}^h={\mathbb{N}}^h\cap \{0,n\},$ .

Also, we make an assumption that ξ is located on the grid, i.e., ξ = mh = m/n, 0 < m < n. We denote ${\mathbb{N}}_o=\{k∊\mathbb{N}:k-odd\},{\mathbb{N}}_e=\{k∊\mathbb{N}:k-even\}.$ .

Let us consider a dSLP (an approximation by Finite-Difference Scheme)

λ ∈ C with classical discrete Dirichlet or Neumann Boundary Condition (BC)

and NBC:

So, we have four cases of BCs: a) (2_d)–(3₀), b) (2_d)–(3₁), c) (2_n)–(3₀), d) (2_n)–(3₁). We denote $\not\asymp =1for\left(3_0\right)BC$ and $\not\asymp =1for\left(3_1\right)BC;K:=gcd(n,m)$ and N := n/K, M := m/K in the case a), b), and $K:=\text{gcd}(\text{2n}-1,\text{2m}-1)∊{\mathbb{N}}_0$ and $K:=\left(\right(\text{2n}-1)/k+1)/2,m:=\left(\right(\text{2m}-1)/k+1)/2$ in the case c).

Let us consider a bijection (see [2])

between ${ℂ}_{\mathrm{\lambda }}:=ℂ$ and ${ℂ}_q^h,{ℂ}_q^h:={ℝ}_q^h\cup {ℂ}_q^{h+}\cup {ℂ}_q^{h-},{ℝ}_q^h:={ℝ}_y^{-}\cup \left\{0\right\}\cup {ℝ}_q^{h+},$

${ℝ}_y^{-}:=\{q=iy:y>0\},{ℝ}_x^h:=\{q=x:0<x<n\}\mathrm{,}{ℝ}_y^{h+}:=\{q=n+iy:\mathrm{y}>0\},$

$C_q^{h+}:=\{q=x+iy:0<x<n,y>0\},C_q^{h-}:=\{q=x+iy:0<x<n,y>0\}$The

general solution $U_j$

Then by using BCs (2) and (3) we get an equation:

where functions Z^h(q) and P^h(q) are determined in Table 1.

Constant Eigenvalues. For any constant eigenvalue λ ∈ Cλ there exists the Constant Eigenvalue Point (CEP) q ∈ Cq. CEP are roots of the system [2]:

For every CEP c_j we define nonregular Spectrum Curve N_j = {c_j}.

Nonconstant eigenvalues. Let us consider Complex Characteristic Function:

All nonconstant eigenvalues (which depend on the parameter γ) are γ-points of (Complex-Real) Characteristic Function (CF)[7]. CF γ(q) is the restriction of Complex CF γc(q) on a set Dξ := {q ∈ ${ℂ}_q^h$ Im γc(q) = 0}. A set Eξ(γ0) := γ−1(γ0) is the set of all nonconstant eigenvalue points for γ0 ∈ R. If q ∈ Dξ and γc′ (q) ̸= 0 (q is not a Critical Point (CP) of CF), then Eξ(γ) is smooth parametric curve N : R → Ch 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]. The regular Spectrum Curves form Spectrum Domain in ${ℂ}_q^h$ ∪ {∞} (see Figure 1).

Each regular Spectrum Curve begins at the pole point (γ = −∞) of CF and ends at the pole point (γ = +∞) of CF. We denote a set Poles $P:=\{p_i,i=\overline{1,n_p}\},$ where np is the number of poles at Ch. For our problems P ⊂ ${ℝ}_x^h$ ∪ {0} and all poles are of the first order (we write deg⁺(p) = 1, p ∈ P).

Two or more Spectrum Curves may intersect at CP. We denote a set $CPB:=\{b_i,i=\overline{1,n_b}\},$, where nb is the number of CPs at ${ℂ}_q^h$. The number of CPs at ${ℝ}_q^h$ and ${ℂ}_q^h$ we denote as ncr and n+ , respectively. Note that the part of the spectrum domain in set ${ℂ}_q^h$ is symmetric to the part in set ${ℂ}_q^h$.$So,n_b=n_{cr}+2n_{cr}^{+}.ifb∊B$ then deg+(b) is one unit larger than the order of CP b.

The pole at q = ∞ is of

order. For (3₁) BC (κ = 1) in the case n = m + 1 the point q = ∞ is CP of the first order. So, $n_{\mathrm{\infty }}^{+}={deg}^{+}\left(\mathrm{\infty }\right)=n_{\mathrm{\infty }}+2\not\asymp \left[\right(m+1)/n].$ A number of all CPs is ${\tilde{n}}_b=n_b+\not\asymp \left[\right(m+1)/n].$

For poles and $CP{deg}^{+}\left(q\right),q\in P\cup B\cup \left\{\mathrm{\infty }\right\}$ corresponds to the number of outgoing Spectrum Curves at that point. Note that incoming Spectrum Curves alternate with outgoing, so deg⁺(q) = deg⁻(q).

2 Graphs. Euler’s characteristic. Digraphs

A graph is a pair of sets G = (V, E) that consists of a non-empty set of vertices (nodes or points), $V=\{v_i,i=\overline{1,I}\in \mathbb{N}\}$ and a set of edges $E=\{e_i,i=\overline{1,J},J\in \mathbb{N}\}$ . We say that $e_j:=(v_{i1},v_{i2})=:v_{i1}{v}_{i2}=v_{i1}{v}_{i2}\in E,v_{i1},v_{i2}\in V,$ and v_i1 (or v_i2 ) is the end of an edge ej. The powers of sets V and E are $\left|V\right|=v$ and $\left|E\right|=e$ The faces of a planar graph are the areas which are surrounded by edges. We denote f the number of such faces.

The Euler’s characteristic χ of a subdivision of a surface is χ = v e + f . Since spectrum curves are on Riemann’s sphere $\overline{ℂ}=ℂ\cup \mathrm{\infty }$, we are interested in Euler’s characteristic of a sphere S2. Euler’s characteristics of a plane [4] of and a sphere [3] are χ = 2.

In graph theory, a directed graph or digraph is a graph that is made up of a set of points connected by arrows (edges with direction). For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex and the number of tail ends adjacent to a vertex is its outdegree. Let G = (V, A) and v ∈ V . The indegree of v is denoted deg⁻(v) and its outdegree is denoted deg⁺(v). If for every verte x v ∈ V , deg⁺(v) = deg⁻(v), the graph is called a balanced digraph. Simple digraphs have no loops and no multiple arrows with same source and target points. The degree sum formula states that, for a digraph,

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.

3 Relations between dSLP and graphs properties

It is possible to define relations between properties of dSLP and graph theory. Poles or CPs refer to vertices of a certain graph and parts of Spectrum Curves could be interpreted as edges. In our case, we have a simple balanced weakly connected digraph.

3.1 Properties of Spectrum Curves

Poles, CPs, regular and nonregular Spectrum Curves, CEPs were found by K. Bingele˙ [1].

There is n -1 Spectrum Curves for every n ∈ N, n > 2. Nonregular Spectrum Curves are CEPs and belong to ${ℝ}_q^h=(0,n).$ The number of such Spectrum Curves is equal to

Number of regular Spectrum Curves n_nce = n − 1 − n_ce. The poles of CF belong to ${ℝ}_q^h$ ∪ {0} ∪ {∞} and n_p + n_∞ = n_nce. So, we have formula

Let us denote

Let n_c is the number of Spectrum Curves parts in ${ℂ}_q^{h+}$ between two CP (including q = ∞ for n = m + 1).

3.2 Spectrum domain as a graph

We consider Spectrum domain as graph on sphere (Riemann sphere $\overline{ℂ}$) because ${ℂ}_q^h\cup \left\{\mathrm{\infty }\right\}~S^2$. The poles and CPs of the CF are the vertices of this graph. The point ∞ is the pole or CP.

Lemma 1. The number of vertices is

From (7) we have .$e={\sum }_{p∊P}de{g}^{+}\left(p\right)+{\sum }_{b∊B}de{g}^{+}\left(b\right)+de{g}^{+}\left(\mathrm{\infty }\right).$ = Σ….

Lemma 2. The number of edges is

Lemma 3. The number of faces is

This lemma is valid for $n_c=n_{cr}^{+}=0$. Each part of spectrum curve between two $CPs{b}_1,b_2\in {ℝ}_q^h$ increases the number of faces by one. So, this formula is valid for the case $n_{cr}^{+}=0.$ Each additional CP b ∈ ${ℂ}_q^{h+}$ increases the number of faces by 2(deg⁺(b) − 1) and number parts of Spectrum Curves between this CP and other CPs by 2 deg⁺(b).

Numbers of spectrum vertices, edges and faces, expressed by the formulas above, inserted to the Euler’s characteristic’s formula of sphere v - e + f = 2 give new relation.

Theorem 1. The Euler’s characteristic’s formula is equivalent to

This formula was derived in [1] when there are no CPs in ${ℂ}_q^{h+\pm }$ $({deg}_r^{+}=2n_{cr})$ all CPs are of the first order $({deg}_r^{+}=2n_{cr})$ and n_c = 0 in the case a), c). Then it can be rewritten as

Corollary 1. The number of edges is

Remark 1. In the case m + 1 < n the formulas (10), (12)–(14) are

where n_ce is defined by (8).

In the case m + 1 = n we have n_∞ = 1 κ and n_p = n 2 + δ, where δ = 1 for the case d) and for n ∈ N_o in the case b), else δ = 0. So, for m + 1 = n the following formulas

are valid.

Remark 2. $n_{cr}^{+}=0({deg}_c^{+}=0)$ then ${deg}_c^{+}{n}_{cr}^{+}={2n}_c+n-m-\not\asymp -1>n_{cr}$ shows that there exist CPs in ${ℝ}_q^h$ of the second or the higher order.

References

[1] K. Bingele˙. Investigation of Spectrum for a Sturm–Liouville problem with Two-Point Nonlocal Boundary Conditions. PhD thesis, Vilniaus Universitetas, 2019.

[2] K. Bingele˙, A. Bankauskiene˙, A. Štikonas. Spectrum curves for a discrete Sturm– Liouville problem with one integral boundary condition. Nonlinear Anal. Model. Control, 24(5):755–774, 2019. https://doi.org/10.15388/NA.2019.5.5.

[3] Peter Hilton, Jean Pedersen. The Euler characteristic and Polya’s dream. American Mathematical Monthly, 103, 02 1996. https://doi.org/10.2307/2975104.

[4] E. Manstavičius. Analizine˙ ir tikimybine˙ kombinatorika. TEV, Vilnius, 2007.

[5] M. Sapagovas. Diferencialiniu˛ lygčiu˛ kraštiniai uždaviniai su nelokaliosiomis sąlygomis. Mokslo aidai, Vilnius, 2007.

[6] A. Skučaite˙. Investigation of the spectrum for Sturm–Liouville problem with a nonlocal integral boundary condition. PhD thesis, Vilnius University, 2016.

[7] A. Štikonas, O. Štikoniene˙. Characteristic functions for Sturm–Liouville prob- lems with nonlocal boundary conditions. Math. Model. Anal., 14(2):229–246, 2009. https://doi.org/10.3846/1392-6292.2009.14.229-246.