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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
Neu osophic Se s and Sys ems, Vol. 97, 2026
Applica ion o neu osophic esol ing se s in ea hquake disas e managemen using
neu osophic g aph models
Shanmugap iya R1,*, Sangee ha P1
1Vel Tech Ranga ajan D . Sakun hala R&D Ins i u e o Science and Technology, Chennai-A adi.
*[email p o ec ed], [email p o ec ed]m
Abs ac :
Neu osophic g aphs a e mo e sui able o modelling eal-li e si ua ions because eal wo ld
da a is o en unce ain, incomple e, inconsis en , o inde e mina e and neu osophic g aphs
a e speci ically designed o handle all o neu osophic g aphs, hese aspec s simul aneously.
In his a icle we in oduced neu o osophic esol ing se , neu o osophic esol ing numbe ,
neu o osophic supe esol ing se , neu o osophic supe esol ing numbe , in e - alued
neu o osophic esol ing se , in e - alued neu o osophic esol ing numbe , also de i ed some
heo ems, p ope ies, co olla ies and also discussed eal li e applica ion based on
neu osophic esol ing se s.
Keywo ds: Neu osophic g aphs, s eng h o connec edness, neu o osophic esol ing
numbe , in e - alued neu o osophic esol ing numbe .
1. In oduc ion
Neu osophic g aphs a e mo e accep able o eal-li e si ua ions because hey allow accu a e,
lexible, and ealis ic modeling o unce ain y, igno ance, and con lic ac o s ha a e inhe en
in nea ly e e y eal-wo ld sys em. Neu osophic g aphs sepa a e he ue, he alse, and he
inde e mina e. In g aph heo y, a esol ing se is a subse o e ices ha uniquely iden i ies
all o he e ices in he g aph based on hei dis ances o he e ices in he se . When ex ended
in o he neu osophic domain, his concep becomes mo e powe ul by inco po a ing u h,
inde e minacy, and alsi y—key elemen s o neu osophic logic— o model unce ain,
incomple e, o inconsis en in o ma ion. A neu osophic esol ing se is a g oup o e ices in
a neu osophic g aph ha allows us o ell apa e e y o he e ex based on he neu osophic
dis ance ec o (which includes u h, inde e minacy, and alsi y) om i o he e ices in he
g oup.
Since i s in oduc ion by Zadeh in 1965 [1], a no el uzzy no ion has been success ully
used o model he di e en unce ain eal-wo ld applica ions in decision-making p oblems.
The uzzy idea is a mo e sophis ica ed e sion o he classical se , wi h dis inc membe ship
alue g ades. The undamen al classical se 's wo u h alues a e ei he 0 o 1. C isp se s a e
inapp op ia e when dealing wi h he unce ain ies o eal-li e p oblems. All i ems in he ype
Uni e si y o New Mexico
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
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1 uzzy se can ha e he p ope membe ship be ween 0 and 1 in he case o 1 o 0. The
an icipa ed esul will esul om his. In his ins ance, he de e mina ion o an objec 's deg ee
o cou se wi hin he uzzy se is cha ac e ized by i s membe ship sco e, which is a unique
numbe ha alls be ween 0 and 1 and is di e en om he p obabili y alue wi hin he uzzy
se .
The indi idual making he choice migh no be capable o managing he unce ain ies
o any complica ed eal-li e si ua ion i hey a e only using one membe ship g ade alue. To
ackle his p oblem, A anasso [2] p esen ed he in ui ionis ic uzzy se (IFS) and i s
cha ac e is ics. Each uzzy se elemen is also assigned a hesi a ion a ing and a non-
membe ship g ade. The uzzy se 's p ope ies can be easily desc ibed by using he h ee
di e en a ibu es and aking in o accoun he pa ame e s, which a e egula ed by IFS
numbe s and include in e io i y, supe io i y, and hesi a ions. Sma andache [3,4] in oduced
he inno a i e idea o neu ophilic se s using he IFS ideology and mo e ele an da a ha
add essed he eal-wo ld issues ela ed o imp ecise, hazy, and unce ain y mo emen . The
neu osophic se is capable o cap u ing he ambigui ies p oduced by i egula , unclea , and
unp edic able da a in any si ua ion. I is essen ially a mo e comple e o m o uzzy se s,
unce ain uzzy se s, and simple adi ional se s alike. Each elemen o he neu osophic se
has been assigned one o h ee membe ship g ades: ambiguous, alse, o ue. The h ee
membe ship classes o he neu osophic se a e independen o each o he and a e always
con ained wi hin [0, 1]. A use ul ool o simula ing eal-wo ld issues is a g aph. Typically,
nodes and loops model he g aph o ep esen he i ems and hei ela ionships. A wide
a ie y o g aph ypes, such as FGs, IFS, and NG heo ies, a e equi ed o ep esen he wide
a ie y o in o ma ion ypes obse ed in p ac ical applica ions [5–11]. IFS ela ionships we e
i s p oposed by Shannon and A anasso [12]. They wen on o publish a numbe o heo ems
and in oduce he concep o in ui ionis ic uzzy g aphs. Pa a hi e al. [13–15] sugges ed a
numbe o di e en me hods o connec wo in ui ionis ic uzzy g aphs. Rashmanlou and
colleagues [16–18] es ablished se e al p oduc ope a ions on IFGs, such as lexicog aphic,
di ec , s ong, and semi-s ong p oduc s. They desc ibe he union on in ui ionis ic uzzy
ne wo ks, he un es ic ed join, and he connec ed componen s. Ac ually, he n-Supe hype
g aph, which was in oduced alongside supe - e ices by Sma andache [19], is he mos
comple e kind o g aph cu en ly accessible. Ak am and colleagues i s p oposed he concep
o a uzzy Py hago ean g aph [20–25]. Acco ding o Khiza Haya e al. [26], he pe manen
unc ion is used as a basis o de e mine he de e minan and adjoin o neu osophic ma ices
wi h in e al alues. Type 2 so g aphs we e de ined by Khiza Haya e al. [27] on unde lying
subg aphs o a simple g aph. Fo neu osophic se s, Fa uk Ka aaslan and Khiza Haya [28]
p o ided e i es ma ices. They p o ided an applica ion o mul i-c i e ia g oup decision-
making based on e i es ma ices. A s udy by Majeed e al. [29] examined se e al index kinds
in neu osophic g aphic ep esen a ions. Bo h deg ee-based and comple ely deg ee-based
indices all wi hin his ca ego y. The neu osophic g aph's e ex absolu e deg ee, -edge
egula , and s ongly edge egula we e in oduced by Ka iya asu, M. [30]. Addi ionally, he
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
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discussed o he aspec s o hese g aphs. The max p oduc o complemen no a ion in NG was
sugges ed by Wadei Fa is AL-Ome i e al. [31] in o de o de e mine he mos e icien web
s eaming pla o ms.
Many o he s ha e explo ed he neu osophic g aph in di e en dimensions (Table 1).
Howe e , no ye aken in o conside a ion is he idea o he sco e unc ion's absolu e alue.
We may concen a e on he issues ha a ise in eal ime du ing ea hquakes in Japan because
he sco e unc ion is c ucial in many decision-making si ua ions. One e en ual goal is o se
up an ea hquake esponse cen e ha aids in he eco e y om such ca as ophes.
Techniques
Sol ed P oblem
Re e ence
Complex in ui ionis ic
FG
Cellula ne wo k p o ide businesses
using uzzy g aphs o es ou me hod
[32]
NG
RSM index modi ica ion
[33]
NG
Find weak edge weigh s
[34]
Colou ing o NG
To de e mine which websi e is phishing
[35]
Pen apa i ioned NG
Finding he sa es ou es
[36]
Complex NG
A chi ec u e o hospi al in as uc u e
[37]
NG
Making decisions and p oposing a
Japanese ea hquake eac ion cen e
[38]
Table 1
Inspi a ion and ex en
• The pu pose o c ea ing a new ma hema ical echnique o da a in eg a ion is o p o ide a
mo e lexible app oach o eal- ime p oblem solu ions.
• By de eloping NG, knowledge is ad anced h ough he use o heo e ical g aphs and
neu osophic collec ions, opening up new a enues o he use o ma hema ical echniques.
• Complex ci cums ances can be add essed mo e eadily by explo ing concep s like union,
join, composi ion o NG, and complica ed homeomo phisms, p o iding aluable in o ma ion
o p oblem sol ing in he eal wo ld.
• Tu key-Sy ia's p oximi y o he Paci ic Ci cle o Fi e makes i ulne able o ea hquakes.
E ec i e seismic esponse cen es mus be se up in o de o mi iga e he e ec s o a disas e .
By accoun ing o he unp edic abili y o in e ac ion, decision-making, and esou ce
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
alloca ion, neu ophilic g aph heo y makes i easie o desc ibe and analyse la ge ne wo ks
ac oss a ange o a eas.
Impo an o his s udy
• Neu ophic collec ions and isualisa ions a e pa icula ly well-sui ed o add ess he
p oblems o ambigui y, inde e minacy, and unce ain y in pa icula con ex s, such as
esponse o ea hquakes o ganising, despi e he ac ha ough se s, uzzy se s, and o he
gene alisa ions o uzzy se s a e impo an and commonly used in many applica ions.
• Disas e esponse cen es in Sy ia and Tu key use MADM models, pa icula ly hose ha
apply neu osophic eason. These models a e impo an because hey p o ide a sys ema ic
amewo k o e alua ing complex choices in he ace o unce ain y, aking s akeholde
p e e ences in o conside a ion, balancing ade-o s, and encou aging adap abili y in
decision-making. Reac ion cen es can imp o e hei managemen and mi iga e he impac o
ea hquakes on a ec ed a eas by pu ing hese s a egies in o p ac ice.
Bene i s and d awbacks
• Compa ed o adi ional c isp o uzzy g aphs, neu ophilic g aphs can be mo e
nuanced in hei ep esen a ion o unce ain y. Decision-make s can mo e accu a ely model
and eason abou unce ain ela ionships when hey a e able o con ey in o ma ion ha is
co ec , e oneous, o unclea . Neu osophic g aphs use u h, inde e minacy, and alsi y
g ading o p o ide a comp ehensi e ep esen a ion o unce ain y. I s g anula i y allows
decision-make s o cap u e e en he smalles a ia ions in he le el o unce ain y, which can
imp o e analysis and decision-making. Analysing neu ophilic g aphs in la ge-scale sys ems
wi h nume ous in e ela ed componen s can be challenging and compu a ionally expensi e.
Since neu ophilic ne wo k opologies necessi a e ce ain expe ise and expe ience, hey may
be challenging o non-expe s o comp ehend o assess.
• The eliabili y and co ec ness o neu onosophic g aph models may be di icul o
e alua e and con i m when he e a e ew g ound u hs o benchma k da ase s. Assessing he
iabili y and e ec i eness o neu osophic g aph-based me hods equi es sensi i i y es s and
s ong alida ion p ocedu es.
The s udy's con ibu ions
Th ough he applica ion o neu osophic g aph heo y o seismic esponse s udy can help
de elop disas e esponse plans ha a e mo e lexible and esilien . Neu ophilic g aphs a e
used o illus a e he complex ne wo k o connec ions, lines o communica ion, and decision-
making p ocesses inside he, enabling mo e comp ehensi e planning and p epa a ion
me hods. This pape 's hemes a e as ollows: NG, union g aphs, sums, complemen s, and
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
g aph composi ions a e de ined in Sec ion 2. We also discuss se e al associa ed p ope ies o
weak and s ong complex NG and desc ibe hei isomo phism. In he hi d sec ion, we desc ibe
he neu osophic esol ing se s on NG and go o e some o he ela ed cha ac e is ics. To
suppo he sugges ed concep s, we p o ide some speci ic examples. The de ini ion and
discussion o he in e al- alued neu osophic esol ing se on in e al- alued NG a e
co e ed in Sec ion 4. Sec ion 5 p esen s a modi ied esol ing se ha is neu osophic,
speci ically applied o modi ied neu osophic g aphs. Conclusions, ecommenda ions, and
applica ions a e included in Sec ion 6.
2. P elimina ies
De ini ion 2.1
Le us assume ha G = ( α, β) and G = ( α, β) a e neu osophic g aphs. An isomo phism ℜ: G
→ G is a map ℜ: V → V ha is bijec i e and ul ils α ( i) = α(ℜ( i)) i V. i.e., Tα( i) =
Tα (ℜ( i )), Iα ( i) = Iα (ℜ( i)), Fα ( i) = Fα (ℜ( i)), i V and β ( i, j) = β (ℜ ( i), ℜ ( j)) ( i,
j) V i.e., Tβ( i, j) = Tβ′ (ℜ ( i), ℜ ( j)), Iβ( i, j ) = Iβ(ℜ( i), ℜ ( j)), Fβ( i, j )= Fβ(ℜ ( i), ℜ ( j)),
( i, j) V. We ep esen ed i as G G.
De ini ion 2.2
Le G = ( α, β) and G = ( α, β) be neu osophic g aphs. The e is a map ℜ : V →V ha sa is ies
β( i, j) = β(ℜ( i), ℜ( j)) ( i, j) V i.e., Tβ( i, j ) = Tβ(ℜ ( i) ℜ ( j)), Iβ( i, j ) = Iβ(ℜ ( i) ℜ
( j)), Fβ( i, j ) = Fβ(ℜ ( i) ℜ ( j)), ( i, j) V. Then ℜ: G → G is a co-weak isomo phism.
De ini ion 2.3
Le G = (α,β) be an SVNG. I G has a pa h P o pa h leng h K. The s eng h o neu osophic
pa h connec ing wo nodes p and q such as P = p = {p1, (p1, p2), p2, ..., pk-1(pk-1, pk)}, pk = q , hen
Tβk(p, q), Iβ
k(p, q) and Fβ
k(p, q) is called he s eng h o he neu osophic pa h. This pa h
desc ibes as ollows.
Tβk (p, q) = sup (Tβ (p, p1) Tβ (p1, p2) ... Tβ(pk-1, pk)),
Iβ
k (p, q) = sup (Iβ (p, p1) Iβ (p1, p2) ... Iβ (pk-1, pk)),
Fβ
k(p, q) = in (Fβ (p, p1) Fβ (p1, p2) ... Fβ (pk-1, pk)).
De ini ion 2.4
Le G = (α,β) be an SVNG. The s eng h o connec ion o a pa h P be ween wo nodes a and b
is de ined by Tβsc(p, q), Iβ
sc(p, q) and Fβ
sc(p, q).
Whe e: Tβsc (p, q) = sup {Tβk(p, q) / k = 1, 2, 3,...}
Iβ
sc (p, q) = sup {Iβ
k(p, q) / k = 1, 2, 3,...}
Fβ
sc (p, q) = in {Fβ
k(p, q) / k = 1, 2, 3,...}.
De ini ion 2.5
Le G [R, S] be an IVFG on a c isp g aph G*(V, E), whe e R = [αlR( i), αuR( j) ] and S =
[αls( i, j) ,αuS( i, j)]. I R is an IVFS on e ex se V and S is an IVFS on edge se E, sa is y
he ollowing condi ion:
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
1) V = { 1, 2, ..., n }, such ha αlR :V→ [0, 1], αuR:V→ [0, 1],
2) αls:V×V→ [0, 1], αuS : V×V→ [0, 1] a e he unc ions ha sa is y ollowing condi ions.
(i) αls( i, j) ≤ min {αuR( i) ,αuR( j)} o all ( i, j)∈ E and
(ii) αus( i, j) ≤ min {αuR( i) ,αuR( j)} o all ( i, j)∈ E.
De ini ion 2.6
Le G (α,β) be IVFG wi h | | = n; a subse o IVFG is
φ = { 1
αl ( 1),αu ( 1), 2
αl ( 2),αu ( 2), 3
αl ( 3),αu ( 3)……… k
αl ( k),αu ( k)},
(α−φ)={ k+1
αl ( k+1),αu ( k+1), k+2
αl ( k+2),αu ( k+2), k+3
αl ( k+3),αu ( k+3)……… n
αl ( n),αu ( n)}.
The way φ is ep esen ed in ela ion o (α- φ) is dis inc , hen he subse φ is said o be an
in e al- alued uzzy esol ing se o G.
De ini ion 2.7
An SVNG wi h e ex se V is de ined by NG
= (α,β), whe e α=(Tα ,Iα ,Fα ) is a single- alued
neu osophic se on VG and β=(T β,Iβ ,Fβ ) is a single- alued neu osophic ela ion on EG
sa is ying he ollowing condi ion:
1) V = { 1, 2,..., n }, such ha Tα :VG → [0, 1], Iα :VG → [0, 1], Fα :VG → [0, 1],
0 ≤ Tα ( i) + Iα ( i) +Fα ( i) ≤ 3, o all i ∈VG .
2) Tβ: EG → [0,1], Iβ: EG → [0,1], and Fβ: EG → [0,1] a e he unc ions ha sa is y he ollowing
condi ions.
i) T β (x,y) ≤min {T α(x),Tα (y)}, (𝑥, y) ∈VG ×VG .
ii) I β (𝑥,𝑦) ≤min {I α(x),Iα (y)}, (𝑥, y) ∈VG ×VG
iii) Fβ (𝑥,𝑦) ≥max {F α(x),Fα (y)}, (𝑥, y) ∈(VG ×VG ) and
0 ≤ Tβ (𝑥,𝑦) + Iβ(x, y) + Fβ(x, y) ≤3 (x,y)∈ E.
3 Neu osophic esol ing se s on neu osophic g aphs
3.1 De ini ion
Le G (α,β) be NG wi h | |= n, a subse o NG is
φ=
{
1
Tα ( 1),Iα ( 1),Fα ( 1),
2
Tα ( 2),Iα ( 2),Fα ( 2),
3
Tα ( 3),Iα ( 3),Fα ( 3),
………..,
k
Tα ( k),Iα ( k),Fα ( k)
}
, (α−φ)=
{
k+1
Tα ( k+1),Iα ( k+1),Fα ( k+1),
k+2
Tα ( k+2),Iα ( k+2),Fα ( k+2),
k+3
Tα ( k+3),Iα ( k+3),Fα ( k+3),
………..,
n
Tα ( n),Iα ( n),Fα ( n)
}
.
The subse 𝜑 is e e ed o as a neu osophic esol ing se o G i i s ep esen a ion 𝜑 in
e e ence o (α- φ) is dis inc . The minimum size o he neu osophic esol ing se is called he
neu osophic esol ing numbe (G).
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3.2 Illus a ion
Figu e.1: Neu osophic g aph
He e V = {a, b, c, d} be he e ex se o G* and E = {ab, bc, cd, da} be he edge se o G*
α = {α1=a
(0.6,0.4,0.60,α2=b
(0.8,0.5,0.5),α3=c
(0.7,0.5,0.4) , α4=d
(0.4,0.6,0.5)} ,
β = {β1=ab
(0.6,0.4,0.6), β2=bc
(0.6,0.5,0.4, β3=cd
(0.4,0.5,0.5), β4=da
(0.4,0.4,0.6)}.
S eng h o connec edness ma ix o NG
Le S1 = {α1, α2}, (α – S) = {α3,α4}
(S1/α3) = {βsc(a,c),βsc(b,c)} = {(0.6, 0.4, 0.6), (0.6, 0.5, 0.4)}
(S1/α4)= {βsc(a,d),βsc(b,d)} = {(0.4, 0.4, 0.6), (04, 0.5, 0.5)}
The ep esen a ion o S1 wi h espec o (α – S1) is dis inc so ha S1 is a esol ing se in NG. In
his same manne S2 = {α1, α3} S3 = {α1, α4}, S4 = {α2, α3}, S5 = {α2, α4} ,S6 = {α3, α4} a e all
esol ing se o G.
3.3 De ini ion
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Le G (α,β) be NG wi h | |= n. A subse φ o NG is said o be supe - esol ing neu osophic se
o G i he ep esen a ion φ wi h espec o α i is dis inc . The minimum size o supe esol ing
neu osophic se is called supe - esol ing neu osophic numbe, deno ed by (G).
3.4 Illus a ion
Figu e 3: Neu osophic g aph
He e e ex se V = { 1, 2, 3, 4, 5}, α={α1, α2,α3,α4, α5} whe e α( i) = αi = ( i
Tα ( i),Iα ( i),Fα ( i))
S eng h o connec edness ma ix o T
S eng h o connec edness ma ix o I
S eng h o connec edness ma ix o F
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
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Le S = { 1, 2} be a supe neu osophic esol ing se o G because he ep esen a ion o s wi h
espec o α is dis inc . So ha (G) = 2.
3.5 Theo em
Neu osophic supe esol ing se is always neu osophic esol ing se bu con e se need no
be ue.
P oo :
Le G be a neu osophic g aph wi h n e ices and le
φ=
{
1
Tα ( 1),Iα ( 1),Fα ( 1),
2
Tα ( 2),Iα ( 2),Fα ( 2),
3
Tα ( 3),Iα ( 3),Fα ( 3),
………..,
k
Tα ( k),Iα ( k),Fα ( k)
}
,
be a neu osophic esol ing se o G. So he ep esen a ion o 𝜑 wi h espec o (α− φ) should
be dis inc bu he se φ ned no be dis inc wi h espec o α so ha neu osophic esol ing se
need no be neu osophic supe esol ing se o G . Con e sely le φ be a neu osophic supe
esol ing se o G hen he ep esen a ion o φ wi h espec o α is dis inc om his
ep esen a ion o φ wi h espec o (α - φ) also hence be neu osophic supe esol ing se is
always be neu osophic esol ing se .
3.6 Theo em
Two neu osophoic g aphs G (V, α, β) and Gˈ(Vˈ, α ˈ,βˈ) a e isomo phic hen 𝓃𝓇(G) = 𝓃𝓇(Gˈ).
P oo :
I G and Gˈ a e isomo phic hen he e exis s a one o one on o mapping R: V →Vˈ such ha
(Tα ( i),Iα ( i),Fα ( i))= (Tαˈ(R ( i)),Iαˈ(R( i)),Fαˈ(R( i)) ∀ i∈V and
[Tβ( i j),Iβ( i j),Fβ( i j)]=[Tβˈ(R ( i j)),Iβˈ(R ( i j)),Fβˈ(R( i j)] ∀ ( i, j)∈V.
Le V= { 1, 2, 3…. n} be e ex se o G and
φ =
{
1
Tα ( 1),Iα ( 1), Fα ( 1),
2
Tα ( 2), Iα ( 2), Fα ( 2),
…………,
p
Tα ( p), Iα ( p), Fα ( p)
}
be a minimal neu osophic esol ing se o G he e o e (G) = p and
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
2) T β: EG → [0, 1], I β:EG → [0, 1], F β:EG → [0, 1] a e he unc ions ha sa is y he ollowing
condi ions.
i) Tβ (𝑥,𝑦) ≤max {Tα(𝑥),Tα (𝑦)}, (𝑥,𝑦) ∈VG × VG .
ii) Iβ (𝑥,𝑦) ≤max {Iα(x),Iα (𝑦)}, (𝑥,𝑦) ∈VG × VG
iii) Fβ (𝑥,𝑦)≥min{Fα(x),Fα (𝑦)}, (𝑥,𝑦) ∈VG × VG and
0 ≤ T β (𝑥,𝑦) + Iβ (𝑥,𝑦) + F β (𝑥,𝑦) ≤3 o all (𝑥,𝑦) ∈ E.
5.3 De ini ion
Le G be a modi ied neu osophic g aph. A p ope subse o is called he modi ied
neu osophic esol ing se o G i he modi ied ep esen a ion o all elemen s in (α−φ) wi h
espec o a e all dis inc . The ca dinali y o he minimum modi ied neu osophic esol ing
se is called he modi ied neu osophic esol ing numbe and is deno ed as (G).
5.4 Illus a ion
Figu e.4: Modi ied Neu osophic g aphs
Weak o connec edness o he abo e neu osophic g aphs is
Le α={q1,q2,q3,q4}, φ1= {q1, q2} so (α−φ1) = {q3, q4}
(φ1/q3)= {[(Twcβ(q1,q3),Iwcβ(q1,q3),Fwcβ(q1,q3)],
[(Twcβ(q2,q3), Iwcβ(q2,q3),Fwcβ(q2,q3)]}= {(.8, .6, .3), (.6, .2, .4)}
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
(φ1/q4) ={[Twcβ(q1,q4),Iwcβ(q1,q4),Fwcβ(q1,q4)]
[Twcβ(q2,q4),Iwcβ(q2,q4),Fwcβ(q2,q4)]}= {(.6, .6, .3), (.8, .5, .3)}
The ep esen a ion o φ1 wi h espec o (α – φ1) a e dis inc so ha φ1 is modi ied esol ing
se in MNG. So ha MNR (G) = 2. In his same manne φ2 = {q1, q3} φ3S3 = {q1, q4}, φ4S4
= {q2, q3}, S5 = {q2, q4}, S6 = {q3, q4} all a e esol ing se o G.
6. Applica ions
The neu osophic g aph is he pe ec model o eal-wo ld si ua ions o disas e
managemen like ea hquakes, loods, and sunamis, whe e in o ma ion is incomple e,
delayed, con using, unce ain, inde e mina e, o pa ially ue. In eme gency managemen ,
a e an ea hquake, eme gency eams need o:
• Please iden i y he impac ed a eas a you ea lies con enience.
• So he places wi h he g ea es deg ee o unce ain y i s .
• Despa ch escue eams op imally.
Howe e :
• Some senso s may be o line.
• Da a om a ec ed a eas may be incomple e ( oads des oyed, communica ion
ailu e).
• Some damage epo s may be con lic ing o delayed.
Thus, unce ain y (inde e minacy) exis s.
Using a neu osophic esolu ion se , we can selec key loca ions (hospi als, eme gency hubs,
and moni o ing cen es) ha bes esol e he unce ain y ac oss all impac ed a eas. Fo he
Tu key – Sy ia Ea hquake 2023 scena io, le us conside a ma hema ical modelling app oach
o ep esen expe opinions on he necessi y and cha ac e is ics o an in e media y measu e
o ea hquake- esis an acili ies in a ious egions. The in o ma ion abou he 2023 Tu key-
Sy ia Ea hquake is displayed in he able 4 below.
A ea
Dea hs
Inju ies
Adana
454.00
7,450.0
Adiyaman
8,387.0
17,499
Ba man
00000
20.000
Diya baki
414.00
902.00
Elazığ
5.0000
379.00
Gazian ep
3,904.0
13,325
Ha ay
24,147
30,762
Kah amanma aş
1,393.0
6,444.0
Kilis
74.000
754.00
Neu osophic Se s and Sys ems, Vol. 97, 2026 373
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
Mala ya
1,393.0
6,444.0
Ma din
1.0000
00000
Osmaniye
1,010.0
2,606.0
Şanlıu a
340
8,919.0
To al
53,537
107,703
Unspeci ied
695.00
8,045.0
Table 4
Le ’s pick 6 eal loca ions a ec ed by he 2023 Tu key–Sy ia ea hquake:
Loca ion
Damage Le el
(app ox.)
Da a Ce ain y
Gazian ep
Hea y
Ce ain
Kah amanma aş
Se e e
Ce ain
Ha ay
Ve y Hea y
Unce ain (delayed epo s)
Adana
Mode a e
Ce ain
Aleppo
Se e e
Unce ain
Mala ya
Hea y
Inde e mina e
(con lic ing epo s)
Table 5
Con e his p oblem in Neu osophic G aph Model
Each ci y is conside ed a e ex. Link ci ies acco ding o hei in as uc u e and geog aphic
p oximi y. The deg ee o ce ain y in connec ions is ep esen ed by edge weigh s.
Fo example:
• Ce ain connec ions: s ong oads, unc ioning communica ion.
• Unce ain connec ions: b oken oads, poo da a low.
Gi e edges and e ices neu osophic elemen s ( alsi y (F), inde e minacy (I), and u h
(T). Reliable s a is ics and p elimina y epo s sugges ha T (ce ain y) is 90%, I
(inde e minacy) is 5%, and F ( alsi y) is 5% in he ci y o Gazian ep. The e is s ong ea hquake
eadiness in Tokyo, acco ding o neu ophilic alues, wi h 80% ag eeing, 20% ha ing a
mode a e a i ude, and 10% disag eeing. One possible ep esen a ion o he Tokyo model is
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
(T: 0.8, 0.2, and 0.1). The eam e alua es expe iewpoin s ega ding he need o es ablish an
ea hquake esponse cen e in each a ea. Wi h us wo hy da a and p elimina y epo s, he
ci y o Gazian ep shows ha T (ce ain y) is 90%, I (inde e minacy) is 5%, and F ( alsi y) is 5%.
A ep esen a ion o Gazian ep would be (0.9, 0.05, 0.05).
Kah amanma aş is he epicen e egion, mos ly e i ied so ha T (ce ain y) is 85%, I
(inde e minacy) is 10%, and F ( alsi y) is 5%. Kah amanma aş may be ep esen ed as (0.85,
0.10, 0.05). The ci y o Ha ay has delayed/con lic ing epo s so ha T (5%), I (35%), and F
(15%), so Ha ay may be ep esen ed as (0.5, 0.35, 0.15). Adana has minimal dis up ion and
eliable senso s, so T (95%), I (3%), and F (2%). The e o e, Adana may be ep esen ed as (0.95,
0.03, 0.02), and Aleppo has poli ical con lic and unce ain da a, so T (60%), I (25%), and F
(15%) o Aleppo may be ep esen ed as (0.6, 0.25, 0.15).
Simila ly Neu osophic Edge Values (connec ions)
Ve ex (Ci y)
T (Ce ain y)
I (Inde e minacy)
F (Falsi y)
Reason
Ci y Gazian ep
0.9
0.05
0.05
Reliable da a, ea ly
epo s
Kah amanma aş
0.85
0.10
0.05
Epicen e egion,
mos ly e i ied
Ha ay
0.5
0.35
0.15
Delayed/con lic ing
epo s
Adana
0.95
0.03
0.02
Minimal dis up ion,
eliable senso s
Aleppo
0.6
0.25
0.15
Poli ical con lic ,
unce ain da a
Mala ya
0.65
0.25
0.10
Repo s con adic ing
se e i y
Table 6
Edges ep esen oad access, communica ion, o da a low. The ci ies o Gazian ep and
Kah amanma aş ha e s ong links and main oads, so T (90%), I (5%), and F (5%). The o e
he neu osophic alues o he edge be ween Gazian ep and Kah amanma aş is (0, 9, 0.05,
0.05) he ci y Gazian ep and Ha ay has Pa ial oad damage, and has delayed da a so T (60%),
I (25%), F (15%) so ha he neu osophic alues o he edge be ween Gazian ep and Ha ay is
(0.6, 0.25, 0.15), he ci y be ween ha ay and Aleppo has Bo de con lic , high unce ain y so
T(40%), I (40%), F (15%) he e o e he edge neu osophic alue be ween ha ay and Aleppo is
(0.4, 0.4, 0.2), he oad be ween Kah amanma aş and Mala ya has blockages so T (70%), I
(20%), F (10%) he e o e he edge neu osophic alue be ween Kah amanma aş and Mala ya
is (0.7, 0.2, 0.1) he ci ies Mala ya and Aleppo a e emo e and unclea ou es so T(30%), I (50%),
F (20%) so ha he edge neu osophic alue be ween Mala ya and Aleppo is (0.3, 0.5, 0.2) he
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
managemen using neu osophic g aph models
ci y Ha ay and Adana has highway in ac so T (85%), I (1%), F (5%) so he edge neu osophic
alue be ween Ha ay and Adana is (0.85, 0.1, 0.15) inally he ci y Aleppo and Adana has
Seconda y oads and da a low uns able so T(60%), I(25%), F(15%) so he edge neu osophic
alue be ween Aleppo and Adana is (0.6, 0.25, 0.15). The ollowing able 7 and igu e 5 show
he neu osophic edge alues.
Edge (Ci y1)
(Ci y2)
T
I
F
No es
Gazian ep
Kah amanma aş
0.9
0.05
0.05
S ong link, main oad
Gazian ep
Ha ay
0.6
0.25
0.15
Pa ial oad damage, delayed da a
Ha ay
Aleppo
0.4
0.4
0.2
Bo de con lic , high unce ain y
Kah amanma aş
Mala ya
0.7
0.2
0.1
Repo s o oad blockages
Mala ya
Aleppo
0.3
0.5
0.2
Remo e, unclea ou es
Ha ay
Aleppo
0.85
0.1
0.05
Highway in ac
Aleppo
Adana
0.6
0.25
0.15
Seconda y oads, da a low uns able
Table 7
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Shanmugap iya R, Sangee ha P, Applica ion o neu osophic esol ing se s in ea hquake disas e
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Figu e. 5
S eng h o connec edness ma ix o T (Ce ain y)
Gazian ep
Kah amanma aş
Mala ya
Ha ay
Aleppo
Adana
Gazian ep
0
0.85
0.65
0.5
0.5
0.5
Kah amanma aş
0.85
0
0.65
0.5
0.6
0.5
Mala ya
0.65
0.65
0
0.5
0.5
0.5
Ha ay
0.5
0.5
0.5
0
0.5
0.5
Aleppo
0.5
0.6
0.5
0.5
0
0.6
Adana
0.5
0.5
0.5
0.5
0.6
0
S eng h o connec edness ma ix o I (Inde e minacy)
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managemen using neu osophic g aph models
Gazian ep
Kah amanma aş
Mala ya
Ha ay
Aleppo
Adana
Gazian ep
0
0.05
0.05
0.05
0.05
0.03
Kah amanma aş
0.05
0
0.1
0.1
0.1
0.03
Mala ya
0.05
0.1
0
0.2
0.25
0.03
Ha ay
0.05
0.1
0.2
0
0.2
0.03
Aleppo
0.05
0.1
0.25
0.2
0
0.03
Adana
0.03
0.03
0.03
0.03
0.03
0
S eng h o connec edness ma ix o F (Falsi y)
Gazian ep
Kah amanma aş
Mala ya
Ha ay
Aleppo
Adana
Gazian ep
0
0.2
0.15
0.15
0.15
0.15
Kah amanma aş
0.2
0
0.1
0.15
0.15
0.15
Mala ya
0.15
0.1
0
0.15
0.15
0.15
Ha ay
0.15
0.15
0.15
0
0.15
0.15
Aleppo
0.15
0.15
0.15
0.15
0
0.15
Adana
0.15
0.15
0.15
0.15
0.15
0
Le us deno e Gazian ep (G), Kah amanma aş (K), Mala ya (M), Ha ay (H), Aleppo (AL),
Adana (A). Le us ake R = {G, K}, (V-R) = {M, H, Al, A}. He e he subse R = {G, K} is a esol ing
se o G.
βSC (M, G), βSC (M, K) = (0.65, 0.05, 0.15), (0.65, 0.1, 0.1)
βSC (H, G), βSC (H, K) = (0.5, 0.05, 0.15), (0.5, 0.1, 0.15)
βSC (AL, G), βSC (AL, K) = (0.5, 0.05, 0.15), (0.6, 0.1, 0.15)
βSC (A, G), βSC (A, K) = (0.5, 0.03, 0.15), (0.5, 0.03, 0.15)
The ep esen a ion o (α - R) wi h espec o R a e all dis inc , he e o e R is he neu osophic
easol ing se o G. E ec i e localiza ion o high-p io i y zones is made possible in seismic
disas e managemen by he use o neu osophic esol ing se s, e en in cases whe e da a is
ambiguous, lacking, o con adic o y. Mo e li es a e e en ually sa ed as a esul o quicke
escue e o s and mo e e ec i e esou ce alloca ion.
Neu osophic esol ing se s bene i s o ea hquake scena ios
Modelling unce ain y po ays ambiguous o con adic o y damage epo s accu a ely.
Imp o ed localiza ion aids in he speci ic iden i ica ion o impac ed a eas o ocused escue
decision suppo helps escue c ews p io i ize asks when da a is lacking. Adap able da a
managemen pe o ms well wi h ambiguous o hazy senso da a (such as ha om d ones o
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managemen using neu osophic g aph models
he in e ne o hings). Managemen o edundancy esol es ambigui ies ha a e no cap u ed
by adi ional g aphs.
6. Conclusions
Neu osophic g aphs a e e ec i e ools o modelling sys ems wi h inconsis en , ambiguous,
and incomple e in o ma ion, which is p e alen in many eal-wo ld domains such as isk
assessmen and disas e managemen , social ne wo k analysis, cybe secu i y and in usion
de ec ion, medical diagnosis sys ems, and decision-making in unce ain en i onmen s. This
manusc ip 's p ima y con ibu ion is he in oduc ion o he concep s o neu osophic
esol ing se s in neu osophic g aphs, neu osophic supe esol ing se s in neu osophic
g aphs, and in e - alued neu osophic esol ing se s in in e - alued neu osophic g aphs.
Addi ionally, we ha e de ined an applica ion based on neu osophic esol ing se s and
discussed a ious heo ems, co olla ies, and p ope ies. We migh in es iga e u he
neu osophic esol ing se p oblems in he u u e. The ea hquake p edic ion cen e has been
es ablished a se e al places in Sy ia and Tu key using he neu osophic g aph. In o de o
make be e decisions, e ices in a neu osophic g aph a e mo e bene icial. In o de o a oid
ca as ophes du ing ea hquakes, he ma hema ical unde pinnings o neu osophic g aph
heo y end o sugges app op ia e places om Tu key o Sy ia.
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Recei ed: Ap il 20, 2025. Accep ed: Sep 30, 2025
