Uni e si y o New Mexico
Mild Balanced Neu osophic G aphs Wi h Applica ion
Kisho e Kuma P.K.1, S. Sangee ha2
1P o esso , Depa men o Ma hema ics, Je usalem College o Enginee ing, Chennai, India;
[email p o ec ed]
2Assis an P o esso , Depa men o Ma hema ics, Je usalem College o Enginee ing, Chennai, India;
[email p o ec ed]
1Co espondence: [email p o ec ed]
Abs ac .The amewo k o mild balanced neu osophic g aphs ex ends adi ional unce ain g aph models
by inco po a ing nuanced membe ship unc ions ha cap u e u h, inde e minacy, and alsi y simul aneously.
This mul i ace ed app oach allows o mo e accu a e ep esen a ion and analysis o unce ain ela ionships in
complex sys ems. The heo e ical ounda ions laid ou h ough igo ous heo ems and p oposi ions p o ide a
obus base o modeling eal-wo ld scena ios whe e ambigui y and pa ial in o ma ion a e inhe en . Mo eo e ,
he applica ion o medicinal in e ac ions in diabe ic pa ien s demons a es he po en ial o hese g aphs in
decision-making p ocesses o op imizing d ug combina ions, he eby enhancing pa ien sa e y and he apeu ic
ou comes. This s udy b idges ma hema ical heo y wi h p ac ical heal hca e challenges, highligh ing he e -
sa ili y and applicabili y o neu osophic g aph models in in e disciplina y esea ch
Keywo ds: Mild balanced neu osophic g aphs, Balanced neu osophic g aphs, Mild Balanced Neu osophic
subg aphs.
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1. In oduc ion
The concep o uzziness in ma hema ical modeling was in oduced by Lo i. A. Zadeh
h ough his g oundb eaking wo ks on uzzy se s [20] and uzzy algo i hms [21]. Sho ly he e-
a e , uzzy g aphs became an ac i e a ea o esea ch, especially ollowing Rosen eld’s wo k
on uzzy ela ionships in cogni i e modeling [16]. Recognizing ha uzziness alone may no
su icien ly cap u e pa ial igno ance o con lic ing in o ma ion, A anasso in oduced in u-
i ionis ic uzzy se s [4], u he o malized in his ex book [5], whe e bo h membe ship and
non-membe ship deg ees a e conside ed simul aneously.
Kisho e Kuma P.K., S. Sangee ha
Neu osophic Se s and Sys ems, Vol. 97, 2026
In ui ionis ic uzzy g aph heo y was u he analysed by Pa a hi and Ka unambigai [14],
who p oposed s uc u ed app oaches o unce ain y ne wo ks. La e ex ensions such as bal-
anced and egula in ui ionis ic uzzy g aphs [9,11], as well as clus e ing algo i hms o knowl-
edge disco e y [10], en iched he ield wi h ools o s uc u e-based analysis unde unce ain y.
In pa allel, neu osophic logic which is p oposed by Flo en in Sma andache inspi ed a mo e
gene alized app oach o modeling ambiguous in o ma ion by inco po a ing h ee componen s:
u h, inde e minacy, and alsi y [18] membe ship unc ions. Kandasamy e al. [8] o malized
neu osophic g aph heo y, enabling highe exp essi eness o complex decision and eason-
ing sys ems. Subsequen s udies explo ed p ope ies o balanced neu osophic g aphs [17].
B oumi e al. [6] in es iga ed on he concep s o single- alued neu osophic g aphs and plana
neu osophic g aphs [12].
Mo eo e , in e al-based and bipola ex ensions we e de eloped o ep esen unce ain da a
a ising om expe assessmen s o con lic ing e idence. Bipola uzzy g aphs, as de ailed
by Ak am e al. [1–3], in oduced a amewo k o modeling sys ems wi h bo h posi i e and
nega i e endencies. A he same ime, in e al- alued neu osophic models [15,19] ha e been
e ec i ely applied in domains like sho es pa h compu a ion, whe e unce ain y anges ma e .
O he ad ancemen s include he inco po a ion o decision suppo indices such as W-
densi y [22] and he RSM index [13] o p edic links in social and knowledge ne wo ks.
Classical con ibu ions such as he E d˝os–R´enyi model o andom g aphs [7] s ill unde pin
much o his wo k, p o iding a p obabilis ic basis o analysing ne wo k p ope ies.
Collec i ely, hese de elopmen s ma k a signi ican shi in g aph modelling echniques om
de e minis ic o imp ecise, om bina y o mul i- alued logic which pa es he way o obus
applica ions in da a sciences, a i icial in elligence, social ne wo ks, and decision suppo sys-
ems. This pape builds upon his ich ounda ion by explo ing new s uc u al and algo i hmic
p ope ies in gene alized uzzy and neu osophic g aphs, con ibu ing o he heo e ical de el-
opmen and p ac ical u ili y o g aph-based unce ain y modelling.
2. P elimina ies
De ini ion 2.1. A single alued neu osophic g aph (SVN-g aph) wi h unde lying se V is
de ined o be a pai G: (A, B) whe e
1. The unc ions TA:V→[0,1], IA:V→[0,1] and FA:V→[0,1] deno e he deg ee
o u h membe ship, deg ee o inde e mina e membe ship and deg ee o alse membe ship
o he elemen i∈V, espec i ely and 0 ≤TA( i) + IA( i) + FA( i)≤3 o all i∈V.
2. The unc ions TB:E⊆V×V→[0,1], ,FB:E⊆V×V→[0,1] a e de ined by
TB( i, j)≤TA( i)∧TA( i), IB( i, j)≤IA( i)∨IA( i) and FB( i, j)∨FA( i)∨FA( i) o
allF( j) which deno es he deg ee o ue, inde e mina e and alse membe ship unc ions o
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 723
he edge ( i, j)∈E espec i ely., whe e 0 ≤TB( i, j) + IB( i, j) + FB( i, j)≤3 o all
( i, j)∈E(i= 1,2, ..., n). We say A, he single alued neu osophic e ex se o V,B he
single alued neu osophic edge se o E.
De ini ion 2.2. A connec ed subg aph H o a neu osophic g aph G: (V, E) is called in ense
subg aph i
(i)V(H)⊆V(G) and E(H)⊆E(G)
(ii) D (H)≤D (G),Di(H)≤Di(G) and D (H)≤D (G)
De ini ion 2.3. A pa ial SVN-subg aph o SVN-g aph G= (A, B) is a SVN-g aph H=
(V′, E′) such ha V′⊆V, whe e T′
A( i)≤TA( i), I′
A( i)≥IA( i), F ′
A( i)≥FA( i) o all
i∈Vand E′⊆E, whe e T′
B( i, j)≤TB( i, j), I′
B( i, j)≥IB( i, j) and F′
B( i, j)≥
FB( i, j) o all ( i, j)∈E.
De ini ion 2.4. Le G: (A, B) be an SVNG. Then, Gis said o be s ong SVNG i TB( j k) =
TA( j)∧TA( k),
IB( j k) = IA( j)∨IA( k) and
FB( j k) = FA( j)∨FA( k).
De ini ion 2.5. Le G: (A, B) be an SVNG. Then, Gis said o be comple e SVNG i
TB( j k) = TA( j)∧TA( k),
IB( j k) = IA( j)∨IA( k) and
FB( j k) = FA( j)∨FA( k).
De ini ion 2.6. A connec ed subg aph H o a neu osophic g aph G: (V, E) is called Feeble
subg aph i
(i)V(H)⊆V(G) and E(H)⊆E(G) and
(ii) D (H)> D (G), Di(H)> Di(G) and D (H)> D (G)
De ini ion 2.7. A neu osophic g aph G: (V, E) is called a mild balanced neu osophic g aph
i all connec ed subg aphs o Ga e in ense subg aphs.
De ini ion 2.8. Two in ense neu osophic connec ed subg aphs H1and H2o a neu osophic
g aph G: (V, E) a e called equally balanced subg aphs i
(i) D (H)≤D (G), Di(H)≤Di(G)and D (H)≤D (G)
(ii) D (H1)≤D (G),Di(H1)≤Di(G) and D (H2)≤D (G)
(iii) D (H1) = D (H2),Di(H1) = Di(H2) and D (H1)≤D (H2)
De ini ion 2.9. I D (Hj) = D (G),Di(Hj) = Di(G) and D (Hj) = D (G) o all possi-
ble connec ed subg aphs Hjo G, hen he g aph G: (V, E) is called a s ic ly balanced
neu osophic g aph.
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 724
3. Mild balanced neu osophic g aphs
Theo em 3.1. Fo a s ong neu osophic g aph, D(G) = (2,2,2) and i is s ic ly balanced.
P oo . Since all he edges o he neu osophic g aph G: (V, E) a e s ong B( j k) = A( j)∧
A( k),
iB( j k) = iA( j)∨iA( k) and
B( j k) = A( j)∨ A( k)
By de ini ion,
Dµ(G) = 2P B( j k)
P A( j)∧ A( k)
=2PiA( j)∧iA( k)
PiA( j)∧iA( k)
=2P A( j)∧ A( k)
P A( j)∧ A( k)
= 2
Hence D(G) = (D (G), Di(G), D (G))b= (2,2,2).
Also all he connec ed subg aphs o G: (V, E) has s ong edges and hence D(H) = (2,2,2)
o all subg aphs Ho G. Hence G: (V, E) is s ic ly balanced.
Co olla y 3.2. A neu osophic g aph wi h ew s ong edges can ne e be mild balanced.
P oo . I a neu osophic g aph G has ew s ong edges(no all he edges), hen o any connec ed
neu osophic subg aph Hwill ha e only s ong edges, D (H) = 2, Di(H) = 2 and D (H) =
2.Hence D(H) = (2,2,2) > D(G). Hence i may no be a mild balanced neu osophic g aph.
Rema k 3.3. I can be no ed ha subg aphs wi h s ong edges a e always eeble subg aphs
o an neu osophic g aph unless i should be s ong.
P oposi ion 3.4. Union o wo equally balanced connec ed neu osophic subg aphs, wi h one
o mo e e ices in common is also equally balanced.
P oo . Le H1and H2be wo equally balanced connec ed neu osophic subg aphs wi h a leas
one common e ex o a neu osophic g aph G: (V, E).
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 725
By de ini ion,
D(H1) = D(H2)≤D(G)
D (H1) = 2P∀ j k∈V(H1) B( j k)
P∀ j k∈E(H1)( A( j)∧ B( k)) =2a
b
D (H2) = 2P∀ j k∈V(H2) B( j k)
P∀ j k∈E(H2)( A( j)∧ B( k)) =2c
d
Since
D (H1) = D (H2) = 2a
b
D (H1∪H2) =
2hP∀ j k∈V(H1) B( j k) + P∀ j k∈V(H2) B( j k)i
P∀ j k∈E(H1)( A( j)∧ B( k)) + P∀ j k∈E(H2)( A( j)∧ B( k))
=2(a+c)
b+d=2(a+ka)
b+kb =2a(k+ 1)
b(k+ 1)
⇒D (H1∪H2) = 2a
b
Hence, D (H1∪H2) = D (H1) = D (H2).
Simila ly we can show ha , Di(H1∪H2) = Di(H1) = Di(H2) and D (H1∪H2) = D (H1) =
D (H2)
⇒D(H1∪H2) = D(H1) = D(H2).
Co olla y 3.5. I all he possible connec ed subg aphs o a mild balanced neu osophic g aphs
a e equally balanced hen he g aph is s ic ly balanced.
P oo . The p oo o his co olla y ollows om di iding he g aph in o wo connec ed subg aphs
which a e balanced equally.
F om he p oposi ion s a ed abo e i ollows ha he union o wo equally balanced connec ed
neu osophic subg aphs is equally balanced, he g aph i sel will be made o a s ic ly balanced
neu osophic g aph.
P oposi ion 3.6. Two connec ed neu osophic g aphs G1and G2wi h a leas one common
e ex a e in ense subg aphs o he neu osophic g aph G1+G2.
P oo . Le G1: (V1, E1) and G2: (V2, E2) be wo connec ed neu osophic g aphs wi h a leas
one common e ex.
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 726
Le
D (H1) = 2P∀ j k∈V(H1) B( j k)
P∀ j k∈E(H1)( A( j)∧ B( k)) =2a
b
D (H2) = 2P∀ j k∈V(H2) B( j k)
P∀ j k∈E(H2)( A( j)∧ B( k)) =2c
d
D (G1+G2) =
2hP∀ j k∈V1,V2 B( j k) + P∀ j k∈V∗ B( j k)i
P∀ j k∈E1,E2( A( j)∧ B( k)) + P∀ j k∈E∗( A( j)∧ B( k))
whe e V∗and E∗a e he se o e ices and s ong edges be ween e e y pai o non-common
e ices G1and G2. The e o e, B( j k) = A(Vi)∧ A( j) o all i j∈E∗.
Since we a e adding a s ong edge be wen all he pai o non-common e ices o G1and G2.
X B( j k) = X A( j)∧ A( k)∀ j k∈E∗
∴D (G1+G2) = 2(a+c+x)
b+d+x>2a
b>2c
d
D (G1+G2)> D (G1) and D(G2)< D(G1+G2).
Simila ly, i can be shown ha
Di(G1+G2)> Di(G1) and D(G2)< D(G1+G2) and
D (G1+G2)> D (G1) and D(G2)< D(G1+G2).
Hence G1and G2a e in ense subg aphs o G1+G2.
In pa icula , D(G1) = D(G1+G2) = D(G2) i all he g aphs a e s ong neu osophic g aphs.
P oposi ion 3.7. Two connec ed neu osophic g aphs G1and G2wi h a leas one common
e ex a e no in ense subg aphs o hei union.(wi hou p oo )
4. Applica ion o Mild balanced Neu osophic g aphs
He e, we apply he ideas o mild balanced neu osophic g aphs in p esc ibing d ugs o sui -
able pa ien s acco dingly. Le us conside he scena io whe e he e a e 4 d ugs which could be
p esc ibed o a diabe ic pa ien . This applica ions de e mines he e iciency o imp o ing he
mix u e o d ugs gi en o a pa ien wi h mul iple illness.
A g aphical ep esen a ion o u h membe ship, inde e mina e membe ship and alse membe -
ship is calcula ed based on medical ea men s and he medicines p esc ibed by he physician.
A pic o ial ep esen a ion o mula ed by hese a e hen embedded in o a neu osophic g aph.
The ollowing d ugs ha e been used by an diabe ic pa ien wi h a ious illness such as hy-
oid, hea disease and blood p essu e. We iden i y he combina ion o d ugs which would be
sui able o an adul o ake i as li e ime e en.
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 727
We ake me o min, empagli lozin, lisinop il and le o hy oxine which a e d ugs used o di-
abe es, ca diac disease, blood p essu e and hy oid espec i ely. In his applica ion we a e
inding a solu ion o he p oblem whe e hese d ugs a e gi en wi h limi ed dose o a pe son
depending on he combina ion which won a ec he pa ien s heal h. The ollowing diag am
shows he in e ac ion diag am o hose d ugs used.
Figu e 1. In e ac ion Diag am o 4 d ugs o a
Diabe ic Pa ien
Figu e 2. Balanced Neu osophic Vague G aph
o 4 D ugs
Kisho e Kuma P.K., S. Sangee ha, Mild Balanced Neu osophic G aphs Wi h Applica ions
Neu osophic Se s and Sys ems, Vol. 97, 2026 728
5. Conclusion
This s udy no only es ablishes a heo e ical ounda ion bu also emphasizes he p ac ical
signi icance o he mild balanced neu osophic g aph model in add essing complex unce ain ies
in di e se domains. The amewo k acili a es be e handling o inde e minacy and incon-
sis ency in ne wo k da a, which is c ucial o eal-wo ld decision-making p ocesses. Looking
ahead, he ex ension o his concep o inco po a e adap i e pa ame e s will enable he mod-
eling o mo e dynamic and he e ogeneous sys ems. Applica ions spanning heal hca e, commu-
nica ion ne wo ks, social in e ac ions, and logis ics s and o bene i om hese ad ancemen s.
Such in eg a ion pa es he way o inno a i e ools ha can p edic and manage unce ain y
wi h enhanced p ecision, ul ima ely imp o ing he e icacy o solu ions in echnology and sci-
ence.
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Recei ed: June 1, 2025. Accep ed: No 18, 2025
