Formulas for Surface Weighted Numbers on Graph

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 6 Issue 3, March-April 2022 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
@ IJTSRD | Unique Paper ID – IJTSRD49573 | Volume – 6 | Issue – 3 | Mar-Apr 2022 Page 784
Formulas for Surface Weighted Numbers on Graph
Ghulam Hazrat Aimal Rasa
Kabul Education University, Kabul, Afghanistan
ABSTRACT
The boundary value problem differential operator on the graph of a
specific structure is discussed in this article. The graph has degree 1
vertices and edges that are linked at one common vertex. The
differential operator expression with real-valued potentials, the
Dirichlet boundary conditions, and the conventional matching
requirements define the boundary value issue. There are a finite
number of eigеnvаluеs in this problem.The residues of the diagonal
elements of the Weyl matrix in the eigenvalues are referred to as
weight numbers. The еigеnvаluеs are monomorphic functions with
simple poles.The weight numbers under consideration generalize the
weight numbers of differential operators on a finite interval, which
are equal to the reciprocals of the squared norms of eigenfunctions.
These numbers, along with the eigеnvаluеs, serve as spectral data for
unique operator reconstruction. The contour integration is used to
obtain formulas for surfacethe weight numbers, as well as formulas
for the sums in the case of superficial near еigеnvаluеs. On the
graphs, the formulas can be utilized to analyze inverse spectral
problems.
KEYWORDS: boundaryproblem, Formulas for Surface, weight
numbers
How to cite this paper: Ghulam Hazrat
Aimal Rasa "Formulas for Surface
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on Graph" Published
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1. INTRODUCTION
Wесconsider the graph Γ which consists of m
edges , 2, 1, ,
j
e m j m
≥ = joined at a common vertex.
We let the graph Γ be parameterized so that
[ ]
0,
j
x π
∈ where the parameter j
x corresponds to the
edge j
e , the parameter 0
j
x = in the boundary vertex
and j
x π
= in the common vertex, 1,
j m
= . We call Γ
a star-shaped graph. A vector function is a graph
function.
1
( ) ( ) ,
m
j j j
y x y x
=
 
=  
Where the components ( )
j j
y x are functions on the
edges
j
e correspondingly [ ]
2
( ) 0, , 1,2,...,
j j
y x L j m
π
∈ = .
Differentiation of the function ( )
g x with respect to
the first parameter is ( )
g x
′ denoted. Consider the
variation in expression.
: ( ) ( ) ( ), 1,...,
j j j j j j
Ly y x p x y x j m
′′
= − + = (1)
The differential operators on the graph for the
boundary value problem can therefore be represented
as follows:
( )
Ly y x
λ
= (2)
1
(0) 0
m
i
i
y
=
=
∑ (3)
1
1
2
( ) ( )
m
j
j
y y
π π
−
=
′ ′
− = ∑ (4)
1 2
( ) ( ) ... ( )
m
y y y
π π π
= = = (5)
where λ is The spectral parameter, the equalities (3),
and the conventional matching criteria (4)–(5) are all
Dirichlet conditions. In (1) the functions ( )
j j
p x are
called potentials, [ ]
2
2
( ) 0, , ( )
j j j j
p x L p x
π
∈ ∈ .The
differential operator L , given by the differential
expression (1) and the conditions (3)–(5), is self-
adjoin in the corresponding Hilbert space (see [1] for
details). The differential operators on graphs are
intensively investigated because they have
applications in physics, chemistry, and
nanotechnology (see [2,3]). We develop formulas for
IJTSRD49573

International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470
Surface weight numbers of the problem (2)–(5) in this
paper which generalize the weight numbers on a finite
interval. The inverse spectral issues for differential
operators on graphs can be studied using these
formulas for Surface. The potentials of the differential
operators on graphs have been reconstructed using
weight numbers and eigenvalues, for example, in
[5,6]. When the eigenvalues are superficial close but
not numerous, the situation becomes more
complicated. The Surface formulas are obtained by
integrating over the contours in the plain of the
spectral parameter that contain the superficial close
eigenvalues. As with the weight matrices for the
matrix differential operator in [7], the Surface
formulas are obtained for the sums of the weight
numbers.
Objectives of this research
The goal of this study is to provide Surface formulas
for the weight numbers of the boundary problem
differential operator on a Star-shaped graph.
Methodology:
On a Star-shaped graph, a descriptive research project
to focus on and discover the effect of differential
equations on Surface for formulae weight numbers of
the boundary problem differential operator. This
research was advanced and completed using books,
journals, and websites.
2. Basic instructions
In this section we introduce a characteristic function of the operator L, the zeros of which coincide with the
eigenvalues. We also provide auxiliary results from [8, 9],related to the eigenvalues of L .
The conditions (4)–(5) can be written as follows:
( ) : ( ) ( ) 0
Y y Hy hy
π π
′ ′
= + =
where H and h are m m
× matrices :
1 1 1 1 1 0 0 0 0 0
0 0 0 0 0 1 1 0 0 0
, 0 1 1 0 0
0 0 0 0 0
0 0 0 0 0 0 0 0 1 1
H h
   
   
−
   
   
= = −
   
   
   
−
   
L L
L L
M M M O M M L
L M M M O M M
L L
For each fixed 1,
j m
= let S ( , )
j x λ and ( , )
j
C x λ be the solutions of the Cauchy problems
S ( , ) ( )S ( , ) S ( , ), S (0, ) S (0, ) 1 0,
j j j j j j
x p x x x
λ λ λ λ λ λ
′′ ′
− + = = − =
( , ) ( ) ( , ) ( , ), (0, ) 1 (0, ) 0.
j j j j j j
C x p x C x C x C C
λ λ λ λ λ λ
′′ ′
− + = − = =
The functionsS ( , )
j x λ , ( , )
j
C x λ satisfy the Volterra integral equations
0
sin sin ( )
S ( , ) ( ) ( , )
x
j j j
x x t
x p t S t dt
λ λ
λ λ
λ λ
−
= + ∫ (6)
0
sin ( )
( , ) cos ( ) ( , )
x
j j j
x t
C x x p t C t dt
λ
λ λ λ
λ
−
= + ∫ (7)
Put : Im
τ λ
= we can obtain the following surface formulas from (6),(7) as :
λ → ∞
0
2
0 0
sin sin ( )
S ( , ) Sin ( )
sin ( ) ( )
sin ( ) ( )sin
x
j j
x t x
j
j
x x t
x t p t dt
x t p t e
t p d dt O
τ
λ λ
λ λ
λ
λ
λ
λ ξ ξ λξ ξ
λ
λ λ
−
= + +
 
−
+ − +  
 
 
∫
∫∫
(8)
0
0 0
cos ( )
S ( , ) cos Sin ( )
cos ( ) ( )
sin ( ) ( )sin
x
j j
x t x
j
j
x t
x x t p t dt
x t p t e
t p d dt O
τ
λ
λ λ λ
λ
λ
λ ξ ξ λξ ξ
λ λ λ
−
′ = + +
 
−
+ − +  
 
 
∫
∫∫
(9)

0
0 0
sin ( )
( , ) cos cos ( )
sin ( ) ( )
sin ( ) ( )cos
x
j j
x t x
j
j
x t
C x x t p t dt
x t p t e
t p d dt O
τ
λ
λ λ λ
λ
λ
λ ξ ξ λξ ξ
λ λ λ
−
= + +
 
−
+ − +  
 
 
∫
∫∫
(10)
0
0 0
( , ) sin cos ( )cos ( )
sin ( ) ( )
sin ( ) ( )cos
x
j j
x t x
j
j
C x x x t t p t dt
x t p t e
t p d dt O
τ
λ λ λ λ λ
λ
λ ξ ξ λξ ξ
λ λ λ
′ = − + − +
 
−
+ − +  
 
 
∫
∫∫
(11)
We introduce matrix solutions of equation (2): { }
( ) ( , ) , 1,2,...,
j j
S diag S x j m
λ λ
= = and
{ }
( ) ( , ) , 1,2,...,
j j
C diag C x j m
λ λ
= = . Every eigenvalue of problem (2)–(5) corresponds to the zero of the
following characteristic function ( )
λ
∆ :
( ): det ( ( ))
Y S
λ λ
∆ = (12)
As ( , )
j
S π λ , ( , )
j
S π λ
′ are entire functions of λ , the function ∆(λ) is also entire. Recon-struttingthe determinant in
(12), we obtain
1 1
( ) ( , ) ( , )
m
m
k j
k j
j k
S S
λ π λ π λ
= =
≠
 
 
′
∆ =
 
 
 
∑ ∏ (13)
Lemma 1. The number 0
λ is an еigеnvаluе of problem (2)–(5) of multiplicity k ifand only if 0
λ is a zero of
characteristic function of multiplicityk . The statement of the Lemma 1 results from the self-adroitness of L and
is proved with the same technique as in [7, Lemma 3]. From the self-adroitness of L it also follows that the
eigenvalues of the boundaryproblem (2)–(5) are real.
Denote 1
0
1
( ) , ( ) ( ).
2
m
j j j
j
w p t dt f z z w
π
=
= = −
∏
∫ Let ( )
, 1, 1
j
z j m
= − be the zeros of ( )
1
( ), .
m j
m
j
w
f z z
m
=
′ = ∑ We
will mean by { } 1
n n
κ
∞
=
different sequences from 2
l .
3. Results obtained
We define and investigate weight numbers based on the Weyl matrix in this paper.Let { } , 1
( ) ( , )
m
jk j j k
x
λ φ λ =
Φ = be
the matrix solution of (2) under the conditions{ } , 1
(0, ) , ( ) 0.
m
jk j k
I Y
φ λ =
= Φ = The matrix { } , 1
( ) (0, )
m
jk j k
M λ φ λ =
′
= − is
called the Weylmatrix and generalize the notion of the Weyl function for differential operators on intervals (see
[4]). Natural spectral characteristics, such as Weyl functions and their generalizations, are frequently employed
for operator reconstruction. A system of 2m columns of the matrix solutions ( ), ( )
C S
λ λ is fundamental, and
one can show, that
( )
1
( ) ( ( )) ( ( ))
M Y S Y C
λ λ λ
−
= (14)
In view of (16) the elements of the matrix { }
, , 1
( ) ( )
m
k l k l
M M
λ λ =
= can be calculated as
,
1
1
( ) ( , ), ( , )
( )
m
k l j l
j
j k
x
M S x C x
π
λ λ λ
λ =
≠
=
′
 
 
=
 
∆  
 
∏ (15)
The elements of the matrix ( )
M λ are monomorphicfunctions, and their poles may be only zeros of the
characteristic function ( )
λ
∆ .Moreover, analogously to [7, Lemma 3], we prove the following lemma:

Lemma 2. If the number 0
λ is a pole of ( )
kl
M λ ,this pole is simple.
Proof. Let 0
λ be a zero of ( )
λ
∆ of multiplicity b. there are exactly b linearly independent
eigenfunctions 1
{ ( )}b
j j j
y x = .corresponding to 0
λ Denote by K such invertible matrix that first b columns of 0
( )
S λ K
are equal to 1
{ ( )}b
j j j
y x = .
If ( ) ( ) ,
X S K
λ λ
= then 1
( ) ( ) ,
S X K
λ λ −
= and [ ]
1
( ) ( ( )) ( ( ))
M K Y X Y C
λ λ λ
−
= .It is sufficient to prove that for any
element of ( )
A λ the number 0
λ cannot be a pole of order greater than 1, where [ ]
1
( ) ( ( )) ( ( )).
A Y X Y C
λ λ λ
−
= If
, 1
( ) { ( )} ,
m
sl s l
A A
λ λ =
= then
The number 0
λ is zero of the numerator of multiplicity not less than 1
b − from that the statement of the theorem
follows.
We introduce the constants ( )
Re ( )
j
n
k
jn kk
s M
λ λ
α λ
=
= which are called weight numbers. We also mean by
1
{ ( )}
n n
z
κ ∞
= different sequences of соntinuоus functions such as:
2
1
max ( )
n
z R
n
z
κ
∞
≤
=
< ∞
∑
where
( )
1,
2 max s
s m
R z
=
= +
The following two theorems summarize the paper's primary findings.
Theorem 1. Let the eigenvalues of L be enumerated as in theorem 1, 1,
k m
= then
2
( )
2
( 1 )
k n
jn
j I n
n
m
m n
κ
α
π
∈
= − +
∑ (16)
2
1
( )
2 (2 )
k n
ms
n
m n
κ
α
π
−
= + (17)
Where
{ }
1
( )
1
( ) min{ : } .
m
s j
n n
j
I n s λ λ
−
=
= =
U
Proof. To prove the theorem, consider ( ) ,
n
z
z n z R
n
λ
π
= + ≤ Substituting ( )
n z
λ λ
= into (8)–(11), we obtain
2 ( )
( 1)
( , ( )) , (2 )
( )
n
n
j n jn jn j j
n
z
S z z q n
n z n
κ
π λ ω ω ω
λ
−  
= − + = −
 
 
)
% % (18)
2 ( )
( , ( )) ( 1) 1
n n
j n
z
S z
n
κ
π λ
 
′ = − +
 
 
(19)
2 ( )
( , ( )) ( 1) 1
n n
j n
z
C z
n
κ
π λ
 
= − +
 
 
(20)
2 ( 1) ( ) ( )
( , ( )) , (2 )
n
n n
j n jn jn j j
z z
C z z q n
n n
λ κ
π λ ω ω ω
−  
′ = − + = −
 
 
)
% % (21)
[ ]
1 2 1 1
det ( ( )), ( ( )),..., ( ( )), ( ( )), ( ( )),..., ( ( ))
( )
det ( ( ))
s l s m
sl
Y X Y X Y X Y C Y X Y X
A
Y X
λ λ λ λ λ λ
λ
λ
− +
=

Where
0
1
( ) ( )cos
2
j j
p l p t ltdt
π
= ∫
% We substitute (18)–(21) into (13), (15) and get
2
1 1
1 1
( )
( 1)
( ( )) ( )
( )
nm m
m
n
n jn
m m
s j
n
j s
z
z z
n z n
κ
λ ω
λ
− −
= =
≠
 
−  
∆ = − +
 
 
 
∑∏ % (22)
2 2
2 2
1 1
,
( )
( 1)
( ( )) ( ( )) ( )
( )
nm m
m
n
kk n n jn
m m
s j
n
s k j s j k
z
M z z z
n z n
κ
λ λ ω
λ
− −
= =
≠ ≠ ≠
 
−  
∆ = − +
 
 
 
∑ ∏ % (23)
Let us denote ( )
1
( ) , ( )
m
n jn
j
f z z r
ω δ
=
= −
∏ % is the circle of center 0 and radius 0
r > .
It can be proved that ( ) ( )
(1), ,
j j
n
z z O n
= + → ∞
% where ( )
, 1,2,3,..., 1
j
n
z j m
= −
% are the zeros of ( )
n
f z
′ if ( ),
z R
δ
∈
then for sufficiently large 2
, ( )
n
n z
λ runs across the simple closed contour, which surrounds
( )
, 1,2,3,..., 1
j
n j m
λ = − Integrating ( ),
kk
M λ after the substitution 2
( )
n z
λ λ
= we have
2
( ) ( )
2 ( )
1
( ( )) .
2
k n
jn kk n
l I n z R
z
M z dz
i n
δ
λ
α λ
π π
∈ ∈
=
∑ ∫
The following formula is obtained from the previous one and (22), (23):
( )
( )
1 1
2
,
1
( )
( ) ( )
1
( )
2 ( )
1
( )
2
m
m
n
jn
s j
s k j s j k
k n
jn m
j
l I n n
z R
n
j
z
z
n
z
dz
z
i m
z z
n
δ
κ
ω
λ
α
κ
π π
= =
≠ ≠ ≠
−
∈ ∈
=
− +
=
− +
∑ ∏
∑ ∫
∏
%
%
(24)
The remainder
( )
n z
n
κ
can be excluded from the denominator of (24) with Taylor expansion as
1
( )
1
min 1
m
j
n
z R
j
z z
−
=
=
− >
∏ % if n is large enough. Besides, 2 2 ( )
( ) 1 ,
n
n
z
z n z R
n
κ
λ
 
= + ≤
 
 
after the designation
( )
( )
1 1
,
1
( )
1
( )
m
m
jn
s j
s k j s j k
kn m
j
n
j
z
g z
z z
ω
= =
≠ ≠ ≠
−
=
−
=
−
∑ ∏
∏
%
%
we get
2
2
( ) ( )
2
( )
2
k n
jn kn
l I n z R
n
g z dz
m i n
δ
κ
α
π
∈ ∈
 
= +
 
 
 
∑ ∫ (25)
We note that ( )
r
δ contains all ( )
, 1,2,3,..., 1
j
n
z j m
= −
% for r R
≥ and large n. Thus,
( ) ( )
( ) ( )
kn kn
z R z r
g z dz g z dz
δ δ
∈ ∈
=
∫ ∫
the numerator of the fraction ( )
kn
g z is a polynomial of degree 2
m − with leading coefficient 1
m− , and its
denominator is a polynomial of degree 1
m− with leading coefficient 1. For ( )
z r
δ
∈ there is the equality
2
1
( ) ( ),
kn
m
g z O r
z
−
−
= + and
1
( )
1
( ) 1 ( ).
2
kn
z r
g z m O r
i δ
π
−
∈
= − +
∫
As r → ∞ we obtain (16). Formula (17) is proved analogously.

Theorem 2.Let ( )
s
z be a zero of ( )
f z
′ of multiplicity ( ) 0, 1
b s t m
> ≤ ≤
Denote { } { }
( ) ( ) ( ) ( )
( ) 1 : , ( ) 1 : ,
s j s j
N s j m z z N s j m z z
′
= ≤ < ≠ = ≤ < = and { }
( )
( ) 1 : s
j
W s j m z ω
= ≤ < ≠ if
( ),
t W s
∈ then
2
ln
( )
2
( )
t
ts n
l N s
n
m
α κ
π
′
∈
= Ω +
∑ (26)
Else
2
ln
( )
2
( )
t
s n
l N s
n
m
α θ κ
π
′
∈
= +
∑ (27)
Where
( ) ( )
1 ( )
( ) 2 ( ) ( ) ( ) ( )
( ) ( )
( ) ( )
, ( )
( ) ( ) ( )
m
s s
j j
j j N s
t s s
s s j s j
j
j N s j N s
z z
b s
z z z z z
ω ω
θ
ω
= ∈
∈ ∈
− −
Ω = − =
− − −
∏ ∏
∏ ∏
and the product over empty set is understood as 1.
Proof. Denote by r such positive number that the circle
( )
s
z z r
− ≤ does notcontain
( )
, ( )
j
z j N s
∈ and
( )
, 0
s
z r R r C
+ < ≥ > we call the circumference of thatcircle ( )
s
γ the following analogue of the formulae (25)
can be proved:
( )
( )
1 1
2
,
1
2 2
( )
( ) ( )
1
m
m
jn
k j
k t j k j t
t n
ln m
j
l N n s
n
j
z
n
dz
m i n
z z
γ
ω
κ
α
π
= =
≠ ≠ ≠
−
′
∈
=
 
−
 
 
= +
 
 
−
 
 
 
∑ ∏
∑ ∫
∏
%
%
(28)
We designate
1 1,
,
( )
1
( )
( )
( )
m
m
n
k j
k t j k j t
t m
j
j
z
F z
z z
ω
= =
≠ ≠ ≠
=
−
=
−
∑ ∏
∏
As j jn n
ω ω κ
− =
% and the coefficients of ( ), ( )
n
f z f z
′ ′ depend on { } { }
1 1
,
m m
j jn
j j
ω ω
= =
% polynomially, we have
1 1,
,
( )
1
( )
( ) ( )
( )
m
m
jn
k j
k t j k j t
t n
m
j
n
j
z
F z z
z z
ω
κ
= =
≠ ≠ ≠
=
−
− =
−
∑ ∏
∏
%
%
(29)
Where ( )
z s
γ
∈ We integrate the fraction ( )
t
F z .
First we consider ( ) 1
b s > . Then ( )
s
z is a zero of ( )
f z of multiplicity ( ) 1
b s + and cardinality of ( )
W s is
( ) 1
m b s
− − in the case when ( )
p W s
∈ the function ( )
p
F z has no pole inside ( ),
s
γ and
2
2
,
p
sn n
n
m
α κ
π
= what is the
same as (26). If ( )
p W s
∉ then

( ) ( ) ( ) ( )
( ) ( )
( ) 1 ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( )
( )
b s b s
s s
j j
k W s
j W s j W s
j k
t b s
s j
j N s
b s z z z z z z
F z
z z z z
ω ω
−
∈
∈ ∈
≠
∈
− − + − −
=
− −
∑
∏ ∏
∏
and
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
s
j
j W s
t s j
s
j N s
b s z
F z dz
z z
γ
ω
∈
∈
−
=
−
∏
∫ ∏
(30)
formula (29) follows from (28)–(30).
Further, let ( ) 1
b s = When ( )
s
z is a zero of ( ),
f z computations are the same as in the case ( ) 1
b s > so we assume
( )
( ) 0,
s
f z ≠ and consequently ( )
p W s
∈ . Rewriting ( )
p
F z as
( )
1
2
( ) 1 ( )
( ) ( )
( ) ( )
t
t t t
f z f z
F z f z
z z z f z
ω ω ω
−
′
 
′
= = −
 
′
− − −
 
and integrating over ( ),
s
γ we obtain (27).
4. Conclusion
This article is divided into three parts as a
consequence of the research. The first section
comprises an introduction, the second section covers
preliminaries, and the third section contains the
proofs of the second and third theorems, as well as
the justification of the approach for extracting two-
point boundary value issues from finite text. On a
Star-shaped graph, a set of eigenvalues of the
asymptotic formula for boundary condition
coefficients and formulas for Surface weight numbers
of the boundary problem differential operator.
Furthermore, the word weight numbers can be stated
to be taken into account. In some years, these are the
leftovers of the Weyl matrix's oblique elements.These
are well-known functions with simple poles that can
only have a limited set of attributes. The assumed
weight numbers were generalized to the weight
numbers of differential operators over a finite time
period, equivalent to the reciprocal of the particular
squared norms. For the unique reconstruction of
operators, these values, coupled with particular
properties, serve as spectral data. We find the
unbalanced duct for the weight numbers using
contour integration, and fours for the values in the
case of closely spaced free. Finally, in graphs,
formulas can be employed to assess inverse spectral.
References
[1] Berkolaiko G., Kuchment P. Introduction to
Quantum Graphs. AMS, Providence, RI, 2013.
370p
[2] Bondarenko N. Spectral analysis for the matrix
Sturm – Liоuvillе operator on a finite interval.
Tamkang J. Math., 2011, vol. 42, no. 3, pp.
305–327. DOI: 10.5556/j.tkjm.42.2011.305-
327.
[3] Freiling G., Yurko V. A. Inverse Sturm –
Liоuvillерrоblеms and their applications.
NewYork, Nova Science, 2001. 305 p.
[4] Ghulam HazratAimal Rasa, АузерханГ. С.,
«GREEN'S FUNCTION UNPERTURBED
BOUNDARY VALUE PROBLEM OF THE
OPERATOR»,Публикациивматериалахмежд
ународныхконференций«Фараби əлемі»2020
[5] Ghulam HazratAimal Rasa, «The Analytical
Nature of the Green's Function in the Vicinity
of a Simple Pole», International Journal of
Trend in Scientific Research and Development
(IJTSRD) Volume 4 Issue 6, September-
October 2020
[6] Hardy G. H., Littlewood J. E., Polya G.
Inequalities. London, Cambridge University
Press, 1934. 456p.
[7] Pivovarchik V. Inverse рrоblеm for the Sturm –
Liоuvillе equation on a star-shaped graph.
Math. Nachr., 2007, vol. 280, no. 1314, pp.
1595–1619. DOI: 10.1002/mana.200410567.
[8] T. Joro and P. Korhonen, Extension of data
envelopment analysis with preference
information, Springer, 2014.

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