Ci a ion: Meenakshi, A.; Kannan, A.;
Mahdal, M.; Ka hik, K.; Gu as, R. A
Compa a i e S udy o Fuzzy
Domina ion and Fuzzy Colo ing in
an Op imal App oach. Ma hema ics
2023,11, 4019. h ps://doi.o g/
10.3390/ma h11184019
Academic Edi o : Kons an in Kozlo
Recei ed: 24 July 2023
Re ised: 10 Sep embe 2023
Accep ed: 11 Sep embe 2023
Published: 21 Sep embe 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ma hema ics
A icle
A Compa a i e S udy o Fuzzy Domina ion and Fuzzy Colo ing
in an Op imal App oach
Annamalai Meenakshi 1, Adhimoolam Kannan 1,2 , Mi osla Mahdal 3, K ishnasamy Ka hik 4,*
and Radek Gu as 3
1Depa men o Ma hema ics, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and Technology,
Chennai 600062, India; [email p o ec ed] (A.M.); [email p o ec ed] (A.K.)
2Depa men o Ma hema ics, Vel Tech Mul i Tech D . Ranga ajan D . Sakun hala Enginee ing College,
Chennai 600062, India
3Depa men o Con ol Sys ems and Ins umen a ion, Facul y o Mechanical Enginee ing, VSB-Technical
Uni e si y o Os a a, 17. Lis opadu 2172/15, 70800 Os a a, Czech Republic;
mi osla [email p o ec ed] (M.M.); [email p o ec ed] (R.G.)
4Depa men o Mechanical Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science and
Technology, Chennai 600062, India
*Co espondence: [email p o ec ed]
Abs ac :
An op imal ne wo k e e s o a compu e o communica ion ne wo k designed, con ig-
u ed, and managed o maximize e iciency, pe o mance, and e ec i eness while minimizing cos
and esou ce u iliza ion. In a ne wo k design and managemen con ex , op imal ypically implies
achie ing he bes possible ou comes be ween a ious ac o s. This esea ch in es iga ed he use o
uzzy g aph edge colo ing o a ious uzzy g aph ope a ions, and i ocused on he e icacy and
e iciency o he uzzy ne wo k p oduc using he minimal spanning ee and he ch oma ic index o
he uzzy ne wo k p oduc . As a ne wo k made o nodes and e ices, measu emen wi h e ices is
a pa ame e o domina ion, and edge measu emen is a pa ame e o edge colo ing, so we used
hese wo pa ame e s in he algo i hm. This pape aims o iden i y an op imal ne wo k ha can
be es ablished using p oduc ou comes. This s udy shows a way o ind an op imal uzzy ne wo k
based on compa a i e op imal pa ame e domina ion and edge colo ing, which can be elabo a ed
wi h applica ions. An algo i hm was gene a ed using an op imal app oach, which was subsequen ly
implemen ed in he o m o applica ions.
Keywo ds: uzzy colo ing; minimum spanning ee; domina ion numbe ; op imal ne wo k
MSC: 05C15; 05C76
1. In oduc ion
A ma hema ical ool known as g aph heo y plays a i al ole in nume ous b anches o
esea ch and echnology. A g aph ypically depic s a eal and ele an p oblem g aphically.
A g aph is a collec ion o se s (K,L), whe e Kis a collec ion o non-emp y e ices, and Lis
an edge se . Kau man (1973) p esen ed he concep o uzzy g aphs, and u he , Rosen eld
(1975) in e p e ed i . Saman a and Pal (2015, 2013) de ined a ious o ms o uzzy g aphs.
In he li e a u e, he e a e many ways o colo g aphs. Fuzzy se heo y and uzzy g aph
heo y ha e made i possible o model mos eal-wo ld si ua ions mo e p ecisely and
adap ably han hei classical coun e pa s [
1
–
7
]. Mo e s udy is being conduc ed on uzzy
g aphs [
8
–
10
]. The usual g aph model o a ne wo k has a collec ion o nodes joined by edges
o connec ions. The ne wo k p o ides a lexible amewo k o loca ing and obse ing
complex sys ems [
11
]. The idea o s udying complex ne wo ks is essen ial and c osses many
academic ields. Real-wo ld p oblems con ain a a ie y o da a ha can be ep esen ed
using a a ie y o g aph ypes, including uzzy g aphs, in ui ionis ic uzzy g aphs, and
Ma hema ics 2023,11, 4019. h ps://doi.o g/10.3390/ma h11184019 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 4019 2 o 16
neu osophic g aphs [
12
–
17
]. A i in oduced he concep s o he so o e se and he so
o e g aph [
18
]. Saman a and o he s p o ided an explana ion o se e al ecen ly c ea ed
ideas abou in ui ionis ic uzzy g aphs (IFGs), as well as a ew key concep s ha had
al eady been es ablished [
19
]. Using he concep s o in ui ionis ic uzzy se s, in ui ionis ic
uzzy ela ions, and index ma ices as a ounda ion, a new gene aliza ion o IFGs has
been p esen ed [
20
]. Many au ho s ha e s udied a ious ypes o domina ions [
21
–
28
] and
de eloped his ield o s udy. The c isp-g aph colo ing echnique has been used o colo he
α
-cu s o hese uzzy g aphs. As a esul , many c isp g aphs a e colo ed o di e en alues
o
α
, and o iden ical uzzy g aphs, he ch oma ic index changes depending on he alue
o
α
[
29
]. Addi ionally, Be sh ein and Bozhenuk sugges ed using he minimax c i e ion
o de ine he ideal cen e alloca ion in uzzy anspo a ion ne wo ks [
30
]. One s udy
de ined he concep o a uzzy g aph and de e mined he minimum numbe o colo s based
on he alue o he sepa a ion deg ee [
31
–
38
]. A new colo ing echnique was employed
o colo a poli ical map and o add ess a b and-new a ic ligh colo ing issue [
39
]. A
colo ed e ex on uzzy g aphs was used o colo maps. To o e come adio equency
issues, Mahapa a e al. ex ended he colo ing app oach o adio uzzy g aphs [
40
–
42
].
The ela ionship (edges) can be mo e meaning ul han he indi idual (nodes) a imes.
Fo ins ance, links a he han nodes a e c ucial in uzzy social ne wo ks. An associa ed
concep known as edge colo ing is c ucial o issues based on unce ain y. Rega ding he
applica ion o g aph colo ing o communica ion sys ems based on u ilizing ing-spli s, in
addi ion o compa ing he peak h oughpu o NoCs o ci culan and mesh opologies
using deadlock- ee ou ing algo i hms, he esul s o high-le el modeling we e p esen ed
in [
43
]. The sugges ed me hod used ewe ha dwa e esou ces while s ill achie ing minimal
ansmission delay and enhanced he mal e iciency [44].
The mo i a ion o his esea ch wo k is o c ea e a new pla o m o iden i y an op imal
ne wo k in o de o de elop an e ec i e op ical ne wo k by u ilizing a ious uzzy g aph
ope a ions based on edge colo and domina ion pa ame e s, including he ope a ions o
esidue p oduc s, symme ic di e ences, max p oduc s, and lexicog aphic p oduc s. Using
compa a i e s udies on domina ion and edge colo ing, ou esea ch and analysis aimed o
assess he ne wo k’s s eng h. We ha e p o ided an algo i hm o examine he e ec i eness
and e iciency o he cons uc ed ne wo ks, which is he main amewo k o his esea ch
inding. Fu he mo e, we ha e de eloped applica ions o he p oduc ope a ion o hese
uzzy ne wo ks.
2. P elimina ies
De ini ion 1
([
12
])
.
A Fuzzy g aph (
H∗FG = (V∗
FG
,
E∗
FG)
) is a pai o unc ions (
σV∗
FG :V∗
FG →[0, 1]
and
µV∗
FG :V∗
FG ×V∗
FG →[0, 1]
) whe e
µV∗
FG (b1
,
b2)≤min{σV∗
FG (b1∗)
,
σV∗
FG (b2∗)}
o b1∗,b2∗∈V∗FG.
De ini ion 2
([
12
])
.
The unde lying g aph o a uzzy g aph is in he o m (
H∗FG = (V∗
FG
,
E∗
FG)
),
whe e
V∗
FG =na1∗∈V∗
FG :σV∗
FG (a1∗)>0o
and
E∗
FG ={(a1∗
,
a2∗)∈V∗
FG ×V∗
FG
:
µV∗
FG
(a1∗,a2∗)>0}.
De ini ion 3
([
12
])
.
A subse (
T∗
F
) o
V∗
FG
is said o be a domina ing se o a uzzy g aph i e e y
e ex in
V∗
FG −T∗
FG
is domina ed by a leas one e ex o
V∗
FG
. The domina ing se (
T∗
FG
) is said
o be minimal i no p ope subse o T∗
FG is a domina ing se .
De ini ion 4
([
12
])
.
An a c (
a1∗ −a2∗
) is said o be s ong i he alue o deg ee o an edge
membe ship o an a c (a1∗ −a2∗) is equal o s eng h o connec edness be ween a1* and a2*.
De ini ion 5 ([12]).A e ex a1* domina es a2* i he e is a s ong a c be ween hem.
Ma hema ics 2023,11, 4019 3 o 16
De ini ion 6
([
34
])
.
A uzzy g aph
ηec =(Vec,σec,µec)
is a se ha is no emp y, oge he
wi h a pai o unc ions
σec :Vec →[0, 1]
and
µec :Vec ×Vec →[0, 1]
, such ha
x
,
yeVec
,
µec(c,d)≤σec(c)Λσec(d)
, whe e
σec(c)
and
µec(c,d)
ep esen he e ex membe ship alues and
he edge membe ship alues, espec i ely.
De ini ion 7
[
35
])
.
A uzzy g aph
ηec =(Vec,σec,µec)
is comple e i
µec(p,q)=min{σec(p)
,
σec(q)} o all p, q ∈Vec, whe e (p, q) ep esen s he edges be ween he e ices p and q.
De ini ion 8
([
35
])
.
A uzzy g aph
ηec =(Vec,σec,µec)
is said o be bipa i e i e ex se
Vec
is di ided
in o wo nonemp y se s
Vec1
and
Vec2
, such ha
µec(Vec1,Vec2)=
0i
ec1
,
ec2eVec1
o
ec1
,
ec2eVec2
.
Fu he , i
µec(Vec1,Vec2)=min{σec( ec1),σec( ec2)}
o all
ec1eVec1and ec2eVec2
, hen
ηec
is called
a uzzy comple e bipa i e g aph.
De ini ion 9 ([29]).Le H = {h1, h2,. . ., hλ}, λ≥1 be he collec ion o neu al hues. Then, uzzy
se (H, k), whe e k: H
→
(0, 1), is known as a collec ion o uzzy colo s, and 0 < k(h
i
)
≤
1; he
colo ’s membe ship alue is he quan i y o each elemen o he combina ion o h
i
wi h he colo
whi e. Hence, he colo (h
i
, k(h
i
)) is e e ed o as he uzzy colo ha ma ches he undamen al colo
h
i
. Thus, he k(h
i
) [
≤
1] amoun o h
i
is mixed wi h 1
−
k(h
i
) o de e mine how much whi e colo is
needed o c ea e he uzzy colo (h
i
, k(h
i
)). As s a ed in he de ini ion abo e, he basic colo is he
building block om which all o he colo s a e c ea ed. Fo example, g een is a undamen al hue. A
“ uzzy g een” colo can be blended o c ea e o he colo s wi h 0.8 uni s o g een and 0.2 uni s o
whi e. This “ uzzy g een” is deno ed by a g een alue o 0.8. Simila ly, ano he uzzy ed colo ( ed,
0.6) may be o med by mixing 0.6 uni s o ed wi h 0.4 uni s o whi e, and so on.
De ini ion 10
([
29
])
.
Le
ηec =(Vec,σec,µec)
be a connec ed uzzy g aph and
cec =cec1,cec2, . . . , ceck
be a se o basic colo s. Now, wo edges a e only gi en wo uzzy
colo s whose basic colo s di e i hey a e adjacen o one ano he ; o he wise, hey may be gi en
uzzy colo s whose basic colo s a e he same. I he colo o any edge is
(ceci
,
ecj(ceci))
, hen
Ceci
is he basic colo o edge
eecj
= (p, q) and
ecj(ceci)
is i s membe ship alue, which is calcula ed as
kecj(ceci)=µec(p.q)
σec(p)∧σec(q)
, whe e
σec(p)and σec(q)
a e he membe ship alues o e ices p and q,
espec i ely. Finally,
µec(p,q)
is he membe ship alue o he edge
eecj
, i.e., (p, q) in he uzzy
g aph ηec.
De ini ion 11
([
29
])
.
The uzzy ch oma ic index o a uzzy g aph is he minimal se o undamen al
colo s equi ed o colo a uzzy g aph. Suppose he e a e M basic hues a he minimal le el, he s eng hs
o edges canno be desc ibed by his ch oma ic index. Fo example, when wo uzzy g aphs ha e iden ical
ch oma ic indices, hese g aphs canno be compa ed using his ch oma ic index. Hence, he e is some
weigh assigned o he ch oma ic index. The weigh is deno ed by Sec, which is de ined by
Sec =∑M
i=1nMax eci(ceci)o
whe e he basic colo
ceci
is used o colo edge
eecj
o some j and he dep h o colo is
keci(ceci)
. Thus,
S is he o al o each basic colo ’s maximum membe ship alues. Now, he ch oma ic index o a uzzy
g aph is deno ed by (M, S), whe e M is he minimum numbe o basic colo s o colo a g aph and
S is i s weigh . We gene ally ollow he ope a ions on uzzy g aph de ini ions om [13].
3. Ope a ions on Fuzzy G aphs Using Edge Colo ing
Mahaba h a e al. [
29
] in oduced De ini ion 11 o de e mine he weigh o colo s in
a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he
o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each colo
used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in e ms
o he sum o he minimum alue o he edge membe ship alues o each colo acco ding
o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and helps
Ma hema ics 2023,11, 4019 4 o 16
us de e mine how e ec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s he
minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M
min
is he
minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such
c isp g aphs is no su icien o de e mine he s eng h o edges, and hence, some weigh
is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he
colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula colo .
The weigh ed minimum o basic colo s used is deno ed as Wmin and is de ined as
Wmin =
M
∑
p=1nmin geq(cp)o
3.1. Residue P oduc o Two Fuzzy G aphs
Le
RF1=(σec1,µec1
) and
RF2=(σec2,µec2)
be wo uzzy g aph ne wo ks o c isp
g aphs
GRF1=(Vec1,Eec1)and GRF2=(Vec2,Eec2)
, espec i ely. Then, i s esidue p oduc
RF1·RF2=(σ1·σ2,µ1·µ2)is de ined as
(i)
∀(a, b) ∈V1×V2, (σ1·σ2)(a,b)=σ1(a)∧σ2(b).
(ii)
∀(a, b) ∈E1and c6=w∈V2,(µ1·µ2)((a,c),(b,w)) =µ1(a,b).
3.1.1. Example
G
1
=RF
1
and G
2
=RF
2
a e he wo uzzy g aphs o
GRF1=(Vec1,Eec1)
and GRF2=(Vec2,Eec2)
, depic ed in Figu es 1and 2, espec i ely. Then, he esidue
p oduc o he uzzy ne wo k is deno ed by RF1·RF2, as shown in Figu e 3.
Ma hema ics 2023, 11, x FOR PEER REVIEW 4 o 20
3. Ope a ions on Fuzzy G aphs Using Edge Colo ing
Mahaba h a e al. [29] in oduced De ini ion 11 o de e mine he weigh o colo s in
a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he
o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each
colo used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in
e ms o he sum o he minimum alue o he edge membe ship alues o each colo ac-
co ding o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and
helps us de e mine how effec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s
he minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M
min
is he
minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such
c isp g aphs is no sufficien o de e mine he s eng h o edges, and hence, some weigh
is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he
colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula
colo . The weigh ed minimum o basic colo s used is deno ed as W
min
and is de ined as
{}

=
=M
pp
c
q
e
gW
1
)(min
min
3.1. Residue P oduc o Two Fuzzy G aphs
Le 𝑅𝐹=(𝜎


,𝜇

) and 𝑅𝐹=(𝜎


,𝜇

) be wo uzzy g aph ne wo ks o c isp
g aphs 𝐺

=𝑉


,𝐸

 and 𝐺

=𝑉


,𝐸

, espec i ely. Then, i s esidue p oduc
𝑅𝐹∙𝑅𝐹
=(𝜎
∙𝜎
,𝜇∙𝜇) is de ined as
(i) ∀ (a, b)
∈
V
1
× V
2
, (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏).
(ii) ∀ (a, b)
∈
E
1
and c  w
∈
V
2
, (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇
(𝑎,𝑏).
3.1.1. Example
G
1
= RF
1
and G
2
= RF
2
a e he wo uzzy g aphs o 𝐺

=𝑉


,𝐸

 and 𝐺

=
𝑉

,𝐸

, depic ed in Figu e 1 and Figu e 2, espec i ely. Then, he esidue p oduc o
he uzzy ne wo k is deno ed by RF
1
·RF
2
, as shown in Figu e 3.
Figu e 1. Fuzzy G aph G
1
.
Figu e 2. Fuzzy G aph G
2
.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF
1
·RF
2
a e calcula ed and shown in Figu e 3.
Figu e 1. Fuzzy G aph G1.
Ma hema ics 2023, 11, x FOR PEER REVIEW 4 o 20
3. Ope a ions on Fuzzy G aphs Using Edge Colo ing
Mahaba h a e al. [29] in oduced De ini ion 11 o de e mine he weigh o colo s in
a uzzy g aph. By ocusing on he op imali y o he uzzy ne wo k, we can compa e he
o mula gi en in De ini ion 11 wi h he sum o he minimal membe ship alue o each
colo used in he uzzy ne wo k. Hence, his esea ch de ines and modi ies he o mula in
e ms o he sum o he minimum alue o he edge membe ship alues o each colo ac-
co ding o he uzzy ne wo k, which yields he op imum alue o he c ea ed ne wo k and
helps us de e mine how effec i e i is. The ch oma ic numbe o a uzzy g aph ep esen s
he minimum numbe o colo s equi ed o colo he g aph. Le us assume ha M
min
is he
minimum numbe o colo s used o colo he g aph. The deg ee o membe ship o such
c isp g aphs is no sufficien o de e mine he s eng h o edges, and hence, some weigh
is associa ed wi h he ch oma ic numbe . These weigh s o he edges can in luence he
colo ing p ocess by indica ing he s eng h o associa ion o an edge wi h a pa icula
colo . The weigh ed minimum o basic colo s used is deno ed as W
min
and is de ined as
{}

=
=M
pp
c
q
e
gW
1
)(min
min
3.1. Residue P oduc o Two Fuzzy G aphs
Le 𝑅𝐹=(𝜎


,𝜇

) and 𝑅𝐹=(𝜎


,𝜇

) be wo uzzy g aph ne wo ks o c isp
g aphs 𝐺

=𝑉


,𝐸

 and 𝐺

=𝑉


,𝐸

, espec i ely. Then, i s esidue p oduc
𝑅𝐹∙𝑅𝐹
=(𝜎
∙𝜎
,𝜇∙𝜇) is de ined as
(i) ∀ (a, b)
∈
V
1
× V
2
, (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏).
(ii) ∀ (a, b)
∈
E
1
and c  w
∈
V
2
, (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇
(𝑎,𝑏).
3.1.1. Example
G
1
= RF
1
and G
2
= RF
2
a e he wo uzzy g aphs o 𝐺

=𝑉


,𝐸

 and 𝐺

=
𝑉

,𝐸

, depic ed in Figu e 1 and Figu e 2, espec i ely. Then, he esidue p oduc o
he uzzy ne wo k is deno ed by RF
1
·RF
2
, as shown in Figu e 3.
Figu e 1. Fuzzy G aph G
1
.
Figu e 2. Fuzzy G aph G
2
.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF
1
·RF
2
a e calcula ed and shown in Figu e 3.
Figu e 2. Fuzzy G aph G2.
Ma hema ics 2023, 11, x FOR PEER REVIEW 5 o 20
Figu e 3. Edge membe ship alue o RF1·RF2.
F om Figu e 3, he minimum numbe o basic colo s used in RF1·RF2 is 4. The weigh
minimum numbe o basic colo s used in cons uc ed esidue p oduc RF1·RF2 = 0.16 + 0.2
+ 0.16 + 0.16 = 0.68. Thus, he Wmin o RF1·RF2 is 0.68.
3.1.2. Find he Weigh o he Minimal Spanning T ee Using K uskal’s Algo i hm
To ind he minimum spanning ee (MST) using he gi en se o edges, we ollow
he ollowing s eps:
Table 1 shows he weigh o he g aph, and Table 2 so s he edges in ascending o de
based on hei weigh . We begin by adding he edge ay-bz wi h a speci ic weigh o he
MST. Nex , we add he edge by-az o he MST wi h a weigh o 0.16. This edge does no
c ea e a cycle wi hin he MST. Mo ing on, we include he edge by-cz wi h a weigh o 0.16
in he MST. This edge also does no c ea e any cycles. We con inue by adding he edge cy-
bz, wi h a weigh o 0.16, o he MST. Once again, his edge main ains he p ope y o no
c ea ing a cycle. Ano he edge, cy-dz, wi h a weigh o 0.2, is added o he MST wi hou
causing any cycles. The edge dy-cz, wi h a weigh o 0.2, is added o he MST, ensu ing
ha no cycles a e o med. We p oceed by adding he edge ax-bzy, weigh 0.28, o he MST.
The inclusion o his edge does no esul in any cycles. The edge cx-by, ha ing a weigh
o 0.28, is included in he MST wi hou c ea ing cycles. Mo ing o wa d, we add he edge
bx-cy wi h a weigh o 0.33 o he MST; no cycles a e in oduced by his inclusion. Simi-
la ly, he edge dx-cy wi h weigh 0.33 is in eg a ed in o he MST wi hou c ea ing any
cycles. Las ly, we come ac oss he edge ax-bz wi h weigh 0.42. Howe e , adding his edge
would c ea e a cycle wi hin he MST. Thus, we disca d i . Ha ing isi ed all he nodes and
ensu ing ha he numbe o edges is ewe han he numbe o nodes, we can conclude
ha he algo i hm can now be s opped.
Table 1. The weigh o a gi en g aph (Figu e 4).
Figu e 3. Edge membe ship alue o RF1·RF2.
Ma hema ics 2023,11, 4019 5 o 16
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed RF
1·
RF
2
a e calcula ed and shown in Figu e 3.
F om Figu e 3, he minimum numbe o basic colo s used in RF
1·
RF
2
is 4. The weigh
minimum numbe o basic colo s used in cons uc ed esidue p oduc RF
1·
RF
2
= 0.16 + 0.2
+ 0.16 + 0.16 = 0.68. Thus, he Wmin o RF1·RF2is 0.68.
3.1.2. Find he Weigh o he Minimal Spanning T ee Using K uskal’s Algo i hm
To ind he minimum spanning ee (MST) using he gi en se o edges, we ollow he
ollowing s eps:
Table 2shows he weigh o he g aph, and Table 1so s he edges in ascending o de
based on hei weigh . We begin by adding he edge ay-bz wi h a speci ic weigh o he
MST. Nex , we add he edge by-az o he MST wi h a weigh o 0.16. This edge does no
c ea e a cycle wi hin he MST. Mo ing on, we include he edge by-cz wi h a weigh o 0.16
in he MST. This edge also does no c ea e any cycles. We con inue by adding he edge
cy-bz, wi h a weigh o 0.16, o he MST. Once again, his edge main ains he p ope y o
no c ea ing a cycle. Ano he edge, cy-dz, wi h a weigh o 0.2, is added o he MST wi hou
causing any cycles. The edge dy-cz, wi h a weigh o 0.2, is added o he MST, ensu ing ha
no cycles a e o med. We p oceed by adding he edge ax-bzy, weigh 0.28, o he MST. The
inclusion o his edge does no esul in any cycles. The edge cx-by, ha ing a weigh o 0.28,
is included in he MST wi hou c ea ing cycles. Mo ing o wa d, we add he edge bx-cy
wi h a weigh o 0.33 o he MST; no cycles a e in oduced by his inclusion. Simila ly, he
edge dx-cy wi h weigh 0.33 is in eg a ed in o he MST wi hou c ea ing any cycles. Las ly,
we come ac oss he edge ax-bz wi h weigh 0.42. Howe e , adding his edge would c ea e
a cycle wi hin he MST. Thus, we disca d i . Ha ing isi ed all he nodes and ensu ing ha
he numbe o edges is ewe han he numbe o nodes, we can conclude ha he algo i hm
can now be s opped.
Table 1. The edges, so ed by weigh in ascending o de .
Edge ay-bz by-az by-cz cy-bz cy-dz dy-cz ax-by cx-by bx-cy dx-cy ax-bz
Weigh 0.16 0.16 0.16 0.16 0.2 0.2 0.28 0.28 0.33 0.33 0.42
Ma hema ics 2023, 11, x FOR PEER REVIEW 6 o 20
Edge ax-by ax-bz bx-cy cx-by dx-cy ay-bz by-az by-cz cy-bz cy-dz dy-cz
Weigh 0.28 0.42 0.33 0.28 0.33 0.16 0.16 0.16 0.16 0.2 0.2
Figu e 4. Minimum Spanning T ee o RF1·RF2.
Table 2. The edges, so ed by weigh in ascending o de .
Edge ay-bz by-az by-cz cy-bz cy-dz dy-cz ax-by cx-by bx-cy dx-cy ax-bz
Weigh 0.16 0.16 0.16 0.16 0.2 0.2 0.28 0.28 0.33 0.33 0.42
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o RF1··RF2 is
shown o be 2.68.
3.1.3. Lowe Domina ion Numbe o RF1.RF2
I is possible o he e o be mo e han one minimal domina ing se in an es ablished
ne wo k. The se wi h he lowes domina ion numbe among all minimal domina ing se s
is conside ed o be he c ea ed ne wo k’s lowes domina ion numbe , which allows us o
es he ne wo k’s op imali y. Le 𝐺 =(𝜎
,𝜇) be he uzzy g aph o he c isp g aph
GDN = (VDN, EDN), and le DN DNDN SSS ,...,, 21 be he minimal domina ing se o GDN. Fi-
nally, le he co esponding domina ing numbe be deno ed by
DNDNDNDN
γ
γ
γ
γ
,...,
3
,
2
,
1. Among his, he lowes ca dinali y o he domina ion num-
be is called lowe domina ion numbe and is deno ed by .
LDN
γ
Le 21212121 .)4(.)3(.)2(.)1( ,, RFRFRFRFRFRFRFRF SandSSS be he sum o he minimal dom-
ina ing se o RF1.RF2 ( e e o Figu e 3)
},,,{
2
.
1
)1( dycybyayS RFRF =; },,,{
2
.
1
)2( dxcxbxaxS RFRF =;
Figu e 4. Minimum Spanning T ee o RF1·RF2.

Ma hema ics 2023,11, 4019 6 o 16
Table 2. The weigh o a gi en g aph (Figu e 4).
Edge ax-by ax-bz bx-cy cx-by dx-cy ay-bz by-az by-cz cy-bz cy-dz dy-cz
Weigh 0.28 0.42 0.33 0.28 0.33 0.16 0.16 0.16 0.16 0.2 0.2
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o RF
1··
RF
2
is
shown o be 2.68.
3.1.3. Lowe Domina ion Numbe o RF1.RF2
I is possible o he e o be mo e han one minimal domina ing se in an es ablished
ne wo k. The se wi h he lowes domina ion numbe among all minimal domina ing se s
is conside ed o be he c ea ed ne wo k’s lowes domina ion numbe , which allows us o
es he ne wo k’s op imali y. Le
GDN =(σDN,µDN
) be he uzzy g aph o he c isp g aph
G
DN
= (V
DN
,E
DN
), and le
SDN1
,
SDN2
,
. . .
,
SDN
be he minimal domina ing se o G
DN
. Fi-
nally, le he co esponding domina ing numbe be deno ed by
γDN1
,
γDN2
,
γDN3
,
. . .
,
γDN
.
Among his, he lowes ca dinali y o he domina ion numbe is called lowe domina ion
numbe and is deno ed by γLDN.
Le
S(1)RF1.RF2
,
S(2)RF1.RF2
,
S(3)RF1.RF2and S(4)RF1.RF2
be he sum o he minimal dom-
ina ing se o RF1.RF2( e e o Figu e 3)
S(1)RF1.RF2={ay,by,cy,dy};S(2)RF1.RF2={ax,bx,cx,dx};
S(3)RF1.RF2={az,bz,cz,dz}
γ(1)RF1.RF2o S(1)RF1.RF2=0.3 +0.6 +0.6 +0.3 =1.8
γ(2)RF1.RF2o S(2)RF1.RF2=2.6.
γ(2)RF1.RF2o S(3)RF1.RF2=2.1.
γLDN−RF1.RF2o RF1RF2is 1.8.
3.2. Symme ic Di e ence o Two Fuzzy G aphs
Le SDF
1
=
(σec1,µec1)
and SDF
2
=
(σec2,µec2)
be wo uzzy g aphs o c isp g aphs
GSDF1=(Vec1,Eec1)andGSDF2=(Vec2,Eec2)
, espec i ely. Then, he symme ic di e ence
be ween SDF
1
and SDF
2
is deno ed by SDF
1⊕
SDF
2
= (
σec1⊕σec2
,
µec1⊕µec2
) and is de ined
as ollows
1. ∀(a,b)∈V1×V2, σSDF1⊕σSDF2(a,b)=σSDF1(a)∧σSDF2(b).
2. ∀a∈V1and (b,c)∈E2(µSDF1⊕µSDF2(a,b),(a,c)=σSDF1(a)∧µSDF2(b,c).
3. ∀a∈V2and (b,c)∈E1(µSDF1⊕µSDF2((b,a),(c,a)) =µSDF1(b,c)∧σSDF2(a).
4. ∀(a,b)/∈E1and (c,w)∈E2
,
(µSDF1⊕µSDF2((a,c),(b,w)) =min{σSDF1(a)
,
σSDF1(b),µSDF2(c,w)}.
5. ∀(a,b)∈E1and (c,w)/∈E2
,
(µSDF1⊕µSDF2((a,c),(b,w)) =min{µSDF1(a,b)
,
σSDF2(c),σSDF2(w)}.
3.2.1. Example
Le G
1
=SDF
1
and G
2
=SDF
2
be he wo uzzy g aphs o c isp g aphs
GSDF1=(Vec1,Eec1)and GSDF2=(Vec2,Eec2), depic ed in Figu es 1and 2, espec i ely.
The symme ic di e ence o uzzy ne wo k SDF1⊕SDF2is shown in Figu e 5.
Ma hema ics 2023,11, 4019 7 o 16
Ma hema ics 2023, 11, x FOR PEER REVIEW 8 o 20
Figu e 5. Edge Membe ship alue o SDF1 ⊕ SDF2.
As shown in Figu e 5, he minimum numbe o basic colo s used in SDF1 ⊕ SDF2 is
7. The weigh minimum numbe o basic colo s used in he cons uc ed symme ic diffe -
ence ne wo k SDF1 ⊕ SDF2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, he Wmin o
SDF1 ⊕ SDF2 is 3.75.
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o SDF1 ⊕ SDF2
was ound o be 7.
3.2.2. Lowe Domina ion Numbe o SDF1 ⊕ SDF2
Le 21
)4(
21
)3(
,
21
)2(
,
21
)1( SDFSDF
Sand
SDFSDF
S
SDFSDF
S
SDFSDF
S⊕⊕⊕⊕ be
some o he minimal domina ing se o SDF1 ⊕ SDF2 (see Figu e 5)
},{
21
)1( ayax
SDFSDF
S=
⊕; },{
21
)2( ayaz
SDFSDF
S=
⊕;
},{
21
)3( dydx
SDFSDF
S=
⊕; }.,{
21
)4( dzdy
SDFSDF
S=
⊕
2
.
1
)1( SDFSDF ⊕
γ
o =
⊕21
)1( SDFSDF
S 2.2
2
.
1
)2( SDFSDF ⊕
γ
o .6.1
21
)2( =
⊕SDFSDF
S
2
.
1
)3( SDFSDF ⊕
γ
o .2.2
21
)3( =
⊕SDFSDF
S
2
.
1
)4( SDFSDF ⊕
γ
o .9.1
21
)4( =
⊕SDFSDF
S
Figu e 5. Edge Membe ship alue o SDF1⊕SDF2.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed
SDF1⊕SDF2a e calcula ed, as shown in Figu e 5.
As shown in Figu e 5, he minimum numbe o basic colo s used in SDF
1⊕
SDF
2
is 7.
The weigh minimum numbe o basic colo s used in he cons uc ed symme ic di e ence
ne wo k SDF
1⊕
SDF
2
= 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, he W
min
o
SDF1⊕SDF2is 3.75.
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o SDF
1⊕
SDF
2
was ound o be 7.
3.2.2. Lowe Domina ion Numbe o SDF1⊕SDF2
Le
S(1)SDF1⊕SDF2
,
S(2)SDF1⊕SDF2
,
S(3)SDF1⊕SDF2and S(4)SDF1⊕SDF2
be some o he mini-
mal domina ing se o SDF1⊕SDF2(see Figu e 5)
S(1)SDF1⊕SDF2={ax,ay};S(2)SDF1⊕SDF2={az,ay};
S(3)SDF1⊕SDF2={dx,dy};S(4)SDF1⊕SDF2={dy,dz}.
γ(1)SDF1.⊕SDF2o S(1)SDF1⊕SDF2=2.2
γ(2)SDF1.⊕SDF2o S(2)SDF1⊕SDF2=1.6.
γ(3)SDF1.⊕SDF2o S(3)SDF1⊕SDF2=2.2.
γ(4)SDF1.⊕SDF2o S(4)SDF1⊕SDF2=1.9.
γLDN−SDF1⊕.SDF2o SDF1⊕SDF2is 1.6.
Ma hema ics 2023,11, 4019 8 o 16
3.3. Max P oduc o Two Fuzzy G aphs
Le MF
1
=
(σm 1,µm 1and MF2=σm 2,µm 2
be wo uzzy ne wo ks o c isp
g aphs
GMF1=Vm 1,Em 1and GMF2=Vm 2,Em 2
, espec i ely. The maximal p od-
uc o uzzy g aphs MF
1
and MF
2
is ep esen ed by MF
1
* MF
2
=
(σm 1
*
σm 2
,
µm 1
*,
µm 2
)
and is de ined as:
(i)
∀(a,b)∈Vm 1×Vm 2,σm 1*σm 2(a,b)=σm 1(a)∨σm 2(b).
(ii)
∀a∈Vm 1and (b,c)∈Em 2,(µm 1*µm 2((a,b),(a,c)) =σm 1(a)∨µm 2(b,c).
(iii)
∀a∈Vm 2and (b,c)∈Em 1,µm 1*µm 2((b,a),(c,a)) =µm 1(b,c)∨σm 2(a).
3.3.1. Example
G
1
=MF
1
and G
2
=MF
2
de ines he wo uzzy g aphs o c isp g aphs
GMF1=(Vec1,Eec1)and GMF2=(Vec2,Eec2)
, depic ed in Figu es 1and 2, espec i ely.
The max p oduc o he uzzy ne wo k MF1* MF2is shown in Figu e 6.
Ma hema ics 2023, 11, x FOR PEER REVIEW 9 o 20
2
.
1SDFSDFLDN ⊕−
γ
o SDF1 ⊕ SDF2 is 1.6.
3.3. Max P oduc o Two Fuzzy G aphs
Le MF1 = (𝜎,𝜇) and 𝑀𝐹=(𝜎
,𝜇) be wo uzzy ne wo ks o c isp g aphs
𝐺=𝑉
,𝐸 and 𝐺=𝑉
,𝐸, espec i ely. The maximal p oduc o uzzy
g aphs MF1 and MF2 is ep esen ed by MF1 * MF2 = (𝜎 * 𝜎, 𝜇*,𝜇) and is de-
ined as:
(i) ∀ (a, b) ∈ 𝑉 × 𝑉,
𝜎∗ 𝜎(𝑎,𝑏)= 𝜎(𝑎) ∨ 𝜎(𝑏).
(ii) ∀ a ∈𝑉 and (b, c) ∈𝐸,
(𝜇∗ 𝜇)((𝑎,𝑏),(𝑎,𝑐))= 𝜎(𝑎) ∨ 𝜇(𝑏,𝑐).
(iii) ∀ a ∈𝑉 and (b, c) ∈𝐸,
𝜇∗ 𝜇(𝑏,𝑎),(𝑐,𝑎)= 𝜇(𝑏,𝑐)∨ 𝜎(𝑎).
3.3.1. Example
G1 = MF1 and G2 = MF2 de ines he wo uzzy g aphs o c isp g aphs 𝐺=
𝑉,𝐸 and 𝐺=𝑉
,𝐸, depic ed in Figu e 1 and Figu e 2, espec i ely. The
max p oduc o he uzzy ne wo k MF1 * MF2 is shown in Figu e 6.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed MF1 * MF2
we e calcula ed, as shown in Figu e 6.
Figu e 6. Edge Membe ship alue o MF1 * MF2.
As shown in Figu e 6, he minimum numbe o basic colo s used in MF1 * MF2 is 4.
The weigh minimum numbe o basic colo s used in he cons uc ed maximal p oduc
ne wo k MF1 * MF2 = 0.37 + 1 + 0.83 + 0.83 = 3.03. Thus, he Wmin o MF1 * MF2 is 3.03.
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o MF1 * MF2 was
ound o be 9.29.
Figu e 6. Edge Membe ship alue o MF1*MF2.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed MF
1
*MF
2
we e calcula ed, as shown in Figu e 6.
As shown in Figu e 6, he minimum numbe o basic colo s used in MF
1
* MF
2
is 4.
The weigh minimum numbe o basic colo s used in he cons uc ed maximal p oduc
ne wo k MF1* MF2= 0.37 + 1 + 0.83 + 0.83 = 3.03. Thus, he Wmin o MF1* MF2is 3.03.
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o MF
1
* MF
2
was
ound o be 9.29.
3.3.2. Lowe Domina ion Numbe o MF1*MF2
Le
S(1)MF1∗MF2
,
S(2)MF1∗MF2
,
S(3)MF1∗MF2and S(4)MF1∗MF2
be some o he minimal dom-
ina ing se o MF1* MF2(see Figu e 6)
S(1)MF1∗MF2={ay,cx,cz,dy}S(2)MF1∗MF2={ay,bx,by,dy};
Ma hema ics 2023,11, 4019 9 o 16
S(3)MF1∗MF2={bx,cx,ay,dy};S(4)MF1∗MF2={ax,bz,cx,dz}.
γ(1)MF1.∗MF2o S(1)MF1∗MF2=6.5;γ(2)MF1.∗MF2o S(2)MF1∗MF2=6.7;
γ(3)SMF1.∗MF2o S(3)MF1∗MF2=7.2;γ(4)MF1.∗MF2o S(4)MF1∗MF2=5.7;
γLDN−MF1.∗MF2o MF1∗MF2is 5.7.
3.4. Lexicog aphic P oduc o Two Fuzzy G aphs
Le LF
1
= (M
1
, P
1
) and LF
2
= (M
2
, P
2
) be wo uzzy g aphs o he c isp g aphs
GLF1=(Vec1,Eec1)and GLF2=(Vec2,Eec2) espec i ely.
The lexicog aphic p oduc o he wo g aphs is deno ed as LF
1·
LF
2
in uzzy g aph pai
(M,P), such ha
(i)
M(a1,b2)=min(M1(a1),M2(b2)),∀(a1,b2)∈M1×M2.
(ii)
P((x,b2)(x,d2)=min(M1(x),M2(b2d2),∀x∈M1,b2d2∈P2.
(iii)
P((a1,b2)(c1,d2)) =min(P1(a1c1),P2(b2d2)),∀a1b1∈P1and b2d2∈P2.
3.4.1. Example
Le G
1
=LF
1
and G
2
=LF
2
be wo uzzy g aphs o c isp g aphs
GLF1=(Vec1,Eec1)
and GLF2=(Vec2,Eec2)
, depic ed in Figu es 1and 2, espec i ely. The lexicog aphic
p oduc o he uzzy ne wo k is deno ed by LF1·LF2, as shown in Figu e 7.
Ma hema ics 2023, 11, x FOR PEER REVIEW 11 o 20
Figu e 7. Edge Membe ship alue o LF1·LF2.
As shown in Figu e 7, he minimum numbe o basic colo s used in LF1·LF2 is 6. The
weigh minimum numbe o basic colo s used in he cons uc ed lexicog aphic p oduc o
ne wo k LF1·LF2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. Thus, he Wmin o LF1·LF2 is 3.
Using K uskal’s algo i hm, he weigh o he minimal spanning ee o LF1·LF2 was
ound o be 5.5.
3.4.2. Lowe Domina ion Numbe o LF1·LF2
Le 2
*
1LFLF
S be he minimal domina ing se (only one domina ing se ) o LF1·LF2
(see Figu e 7)
}.,,,{
2
*
1
dycybyay
LFLF
S=2
*.
1LFLFLDN −
γ
o .3
2
*
1is
LFLF
S
Figu e 7. Edge Membe ship alue o LF1·LF2.
Using De ini ion 10, he edge membe ship alues o he abo e cons uc ed LF
1·
LF
2
we e calcula ed, as shown in Figu e 7.
As shown in Figu e 7, he minimum numbe o basic colo s used in LF
1·
LF
2
is 6. The
weigh minimum numbe o basic colo s used in he cons uc ed lexicog aphic p oduc o
ne wo k LF1·LF2= 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. Thus, he Wmin o LF1·LF2is 3.
Ma hema ics 2023,11, 4019 16 o 16
13.
Meenakshi, A.; My h eyi, O.; B amila, M.; Kannan, A.; Senbagamala , J. Applica ion o neu osophic op imal ne wo k using
ope a ions. J. In ell. Fuzzy Sys . 2023,45, 421–433. [C ossRe ]
14.
Yang, H.L.; Guo, Z.L.; She, Y.; Liao, X. On single alued neu osophic ela ions. J. In ell. Fuzzy Sys .
2016
,30, 1045–1056. [C ossRe ]
15. A anasso , K.T. On In ui ionis ic Fuzzy Se s Theo y; Sp inge : Be lin, Ge many, 2012.
16.
B oumi, S.; Sma andache, F. New Dis ance and Simila i y Measu es o In e al Neu osophic Se s. In P oceedings o he 17 h
In e na ional Con e ence on In o ma ion Fusion, Salamanca, Spain, 7–10 July 2014.
17.
Islam, S.R.; Pal, M. Hype -Wiene index o uzzy g aph and i s applica ion in sha e ma ke . J. In ell. Fuzzy Sys .
2021
,
41, 2073–2083. [C ossRe ]
18. A i , N.E. Domina ion (se and Numbe ) in Neu osophic So o e G aphs. Wasi J. Pu e Sci. 2022,1, 26–43.
19.
Rashmanlou, H.; Saman a, S.; Pal, M.; Bo zooei, R.A. In ui ionis ic uzzy g aphs wi h ca ego ical p ope ies. Fuzzy In . Eng.
2015
,
7, 317–334. [C ossRe ]
20. Shannon, A.; A anasso , K. On a gene aliza ion o in ui ionis ic uzzy g aphs. NIFS 2006,12, 24–29.
21. Bondy, J.A.; Mu y, U.S.R. G aph Theo y wi h Applica ions; Else ie : New Yo k, NY, USA, 1982.
22. O e, O. Theo y o G aphs; Ame ican Ma hema ical Socie y Colloquium Publica ions: P o idence, RI, USA, 1962.
23.
Haynes, T.W.; Hede niemi, S.; Sla e , P. Fundamen als o in Domina ion in G aphs; Sp inge In e na ional Publishing: New Yo k, NY,
USA, 1998.
24. Swamina han, V.; Dha malingam, K.M. Deg ee equi able domina ion on g aphs. K aguje . J. Ma h. 2011,35, 191–197.
25. Meenakshi, A. Equi able domina ion o complemen o in la ed g aph. AIP Con . P oc. 2019,2112, 1–6.
26. Meenakshi, A.; Babujee, J.B. Pai ed Equi able domina ion in g aphs. In . J. Pu e Appl. Ma h. 2016,109, 75–81.
27. Meenakshi, A. Pai ed Equi able domina ion in in la ed g aph. In . J. Inno . Technol. Explo . Eng. 2019,8, 9–16.
28.
Meenakshi, A.; My h eyi, O. Applica ions o Neu osophic social ne wo k using max p oduc ne wo ks. J. In ell. Fuzzy Sys .
2023
,
45, 407–420. [C ossRe ]
29. Mahapa a, R.; Saman a, S.; Pal, M. Applica ions o edge colou ing o uzzy g aphs. In o ma ica 2020,31, 313–330. [C ossRe ]
30.
Be sh ein, L.; Bozhenyuk, A.; Rozenbe g, I. Op imum alloca ion o cen e s in anspo a ion ne wo ks using uzzy g aph bases. In
P oceedings o he 8 h Con e ence o he Eu opean Socie y o Fuzzy Logic and Technology (EUSFLAT-13), Milano, I aly, 11–13
Sep embe 2013; pp. 270–275.
31.
Mahapa a, T.; Gho ai, G.; Pal, M. Fuzzy ac ional colo ing o uzzy g aph wi h i s applica ion. J. Ambien . In ell. Humaniz.
Compu . 2020,11, 5771–5784. [C ossRe ]
32.
Mahapa a, R.; Saman a, S.; Pal, M.; Lee, J.G.; Khan, S.K.; Naseem, U.; Bhado ia, R.S. Colou ing o COVID-19 a ec ed egion
based on uzzy di ec ed g aphs. Compu . Ma e . Con in. 2021,68, 1219–1233. [C ossRe ]
33. Lewis, R. A Guide o G aph Colou ing; Sp inge : London, UK, 2016.
34. Malagu i, E.; To h, P. A su ey on e ex colo ing p oblems. In . T ans. Ope . Res. 2010,17, 1–34. [C ossRe ]
35.
Galinie , P.; Hamiez, J.P.; Hao, J.K.; Po umbel, D. Recen ad ances in g aph e ex colo ing. In Handbook o Op imiza ion: F om
Classical o Mode n App oach; Sp inge : Be lin, Ge many, 2013; pp. 505–528.
36. Lih, K.W. The equi able colo ing o g aphs. Handb. Comb. Op im. 1998,1, 2015–2038.
37. Leigh on, F.T. A g aph colo ing algo i hm o la ge scheduling p oblems. J. Res. Na l. Bu . S and. 1979,84, 489. [C ossRe ]
38. Fu manczyk, H.; Kubale, M. Equi able colo ing o g aphs. Con emp. Ma h. 2004,352, 35–54.
39. Saman a, S.; P amanik, T.; Pal, M. Fuzzy colou ing o uzzy g aphs. A . Ma . 2016,27, 37–50. [C ossRe ]
40.
Chang, G.J.; Chen, S.H.; Hsu, C.Y.; Hung, C.M.; Lai, H.L. S ong edge-colo ing o jelly ish g aphs. Disc e . Ma h.
2015
,
338, 2348–2355. [C ossRe ]
41.
Kisho e, A.; Suni ha, M.S. On Injec i e Colo ing o G aphs and Ch oma ici y o Fuzzy G aphs; LAP Lambe Academic Publishing:
Saa b ücken, Ge many, 2016.
42.
Rosyida, I.; Ind a i, C.R.; Sugeng, K.A. A new app oach o de e mining uzzy ch oma ic numbe o uzzy g aph. J. In ell. Fuzzy
Sys . 2015,28, 2331–2341. [C ossRe ]
43.
Romano , A.Y.; Myachin, N.M.; Lezhne , E.V.; I anniko , A.D.; El-Mesady, A. Ring-Spli : Deadlock-F ee Rou ing Algo i hm o
Ci culan Ne wo ks-on-Chip. Mic omachines 2023,14, 141. [C ossRe ] [PubMed]
44.
Li, Z.; Shen, R.; Yi, M.; Song, Y.; Wang, X.; Du, G.; Huang, Z. Ho spo s Reduc ion o GALS NoC Using a Low-La ency Mul is age
Packe Reo de ing App oach. Mic omachines 2023,14, 444. [C ossRe ]
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