Neu osophic Se s and Sys ems, Vol. 96, 2026
Uni e si y o New Mexico
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D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
D.Rajalaxmi1, V.Vijaya2and S.Re a hi3
1,2PG and Resea ch Depa men o Ma hema ics, See halakshmi Ramaswami College,
Ti uchi appalli,(A ilia ed o Bha a hidasan Uni e si y, Ti uchi appalli),
[email p o ec ed], [email p o ec ed]
3Depa men o Ma hema ics, Sa ana han College o Enginee ing, Ti uchi appalli, India
[email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : This pape in oduces an e icien algo i hm o de ec ing b idges in Bijec i e
neu osophic g aphs, an eme ging app oach in handling unce ain y in complex sys ems such
as medical diagnos ics. The algo i hm iden i ies T-b idges, I- b idges and F-b idges in
neu osophic g aphs, and inco po a es a de neu osophica ion me hod using sco e unc ion o
ind neu osophic b idges when needed. The applica ion o he algo i hm o cance diagnosis
is explo ed, demons a ing i s po en ial o enhance disease iden i ica ion and ea men
planning based on symp oms. The esul sugges ha his me hod can imp o e he accu acy
o medical decision- making in unce ain en i onmen s, he eby o e ing a p omising ool o
heal hca e analysis. By p o iding a mo e p ecise unde s anding o medical da a, his
app oach has he po en ial o op imize diagnos ic p ocesses and ea men s a egies in
unce ain and inde e mina e con ex s.
Keywo ds: Neu osophic G aph; Bijec i e Neu osophic g aph; T-b idge, I-b idge, F-
b idge; and Neu osophic B idge.
1. In oduc ion
G aph heo y has eme ged as a powe ul ma hema ical ool o modeling complex
sys ems in a ious domains, including biology, compu e science, and medicine. Among i s
nume ous applica ions, he analysis o g aph s uc u es o e s aluable insigh s in o he
connec i i y and c i ical componen s o ne wo ks. In his con ex , b idge de ec ion plays a
signi ican ole in iden i ying he mos sensi i e o pi o al connec ions whose emo al may
Neu osophic Se s and Sys ems, Vol. 96, 2026 285
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D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
agmen he g aph. To add ess he limi a ions o classical g aph models in handling
unce ain y, incomple eness and inconsis ency inhe en in eal wo ld da a- pa icula ly in
medical diagnos ics- he concep o neu osophic g aphs has gained ac ion. These g aphs
ex end uzzy and in ui ionis ic uzzy models by inco po a ing deg ees o u h,
inde e minacy, and alsi y. A u he ad ancemen , bijec i e neu osophic g aphs, ensu es a
one o one co espondence be ween g aph elemen s, o e ing a mo e p ecise and s uc u ed
ep esen a ion o da a. Ak am, M e . al [1] in oduced he concep o ope a ions on Single
Valued Neu osophic G aphs (SVN-g aphs) in 2017 Beaula Thanga aj e al. [2–4]
con ibu ed signi ican ly o he ield by applying a ious uzzy numbe s and anking me hods
o sol e c i ical pa h p oblems, exempli ying he expanding applica ion o uzzy logic in
op imiza ion and decision-making. I is impo an o acknowledge B oumi, S e al[5-6]
s udies abou single alued neu osophic g aph in 2016. Fu he his concep ake i s shape
as neu osophic labelling g aph in 2019, which was in oduced by Goma hi e .al [7].
A.Hassan e .al [8] s udied abou single alued ees in 2018. Mu hu aj e .al[9] s udied abou
mul i uzzy g aph in 2020. Rajalaxmi D e .al[10-11] s udied abou me ic in uzzy labeling
g aph and Bijec i e single alued in highligh ing hei s uc u al p ope ies and po en ial
applica ions. Vijaya e al[12,13] ha e shaped he ad ancemen s o Neu osophic g aphs in
ind he solu ion o Decision making p oblem and c i ical pa h p oblems by using
Py hago ean Fuzzy numbe s and Neu osophic Fuzzy numbe s. Ye, J.,[14-15]s udied Single-
Valued Neu osophic Minimum Spanning T ee in 2014. Finally, he pionee ing wo k o
Zadeh, L. [16] in 1965, who in oduced he concep o uzzy se s, laid he g oundwo k o
he en i e ield o uzzy and neu osophic ma hema ics ha ollowed.
An algo i hmic me hod o de ec ing b idges in bijec i e neu osophic g aphs is
p esen ed in his a icle, wi h a ocus on i s use in he diagnosis o cance . In addi ion o
imp o ing s uc u al analysis o in ica e ne wo ks, he algo i hm o e s a use ul ool ha can
be applied o a ange o eal-wo ld issues. The sugges ed echniques seek o assis oncology's
ea ly de ec ion, isk assessmen , and s a egic in e en ion planning by loca ing impo an
pa hways o in e ac ions in biological ne wo ks linked o cance . Th ough his s udy, we
show how algo i hmic app oaches and ma hema ical abs ac ion can g ea ly ad ance medical
esea ch and decision-making.
The neu osophic amewo k pe mi s he simul aneous ep esen a ion o u h,
inde e minacy, and alsi y, he neu osophic amewo k is especially use ul in his si ua ion.
Incomple e, ambiguous, o con adic ing in o ma ion is equen ly p esen in biological and
medical ne wo ks, pa icula ly hose pe aining o he diagnosis o cance . The abili y o
uzzy g aph models and e en classical g aph heo y o handle such unce ain ies is
cons ained. The sugges ed app oach mo e success ully cap u es se e al aspec s o
unce ain y by using bijec i e neu osophic g aphs, p oducing obus analysis and mo e
us wo hy esul s. This makes he neu osophic app oach especially impo an in ad ancing
medical esea ch and decision-making p ocesses whe e ambigui y is inhe en .
2. P elimina ies:
Neu osophic Se s and Sys ems, Vol. 96, 2026 286
De ini ion 2.1[7]: A neu osophic g aph is o he o m G = (V, σ,μ ) whe e σ = (T1, I1, F1)
and μ = (T2, I2, F2) (i) V = { 1, 2, 3, ···, n} such ha T1: V → [0, 1], I1: V → [0, 1] and
F1 : V → [0, 1] deno e he deg ee o u h-membe ship unc ion, inde e minacy-membe ship
unc ion and alsi y-membe ship unc ion o he e ex i ∈ V espec i ely, and 0 ≤ T1 ( ) +
I1 ( ) + F1 ( ) ≤ 3 ∀ i ∈ V (i=1, 2, 3….n).
(ii) T2 : V × V → [0, 1], I2 : V × V → [0, 1] and F2 : V × V → [0, 1], whe e T2( i, j) ,
I2( i, j) and F2( i, j) deno e he deg ee o u h-membe ship unc ion, inde e minacy
membe ship unc ion and alsi y-membe ship unc ion o he edge ( i, j) espec i ely such
ha o e e y ( i, j), T2 ( i, j) ≤ min {T1( i), T1( j)}, I2 ( i, j) ≤ min {I1( i), I1( j
)},
F2 ( i, j) ≤ max {F1 ( i), F1( j)}, and 0 ≤ T2( i, j) + I2( i, j) + F2( i, j) ≤ 3 .
De ini ion 2.2
A neu osophic g aph is said o be a bijec i e neu osophic g aph i
]1,0[:],1,0[:],1,0[:],1,0[:],1,0[:],1,0[: →→→→→→ VVVVVVVVV FITFIT
a e bijec i e, such ha he u h- membe ship unc ion, Inde e minacy-membe ship unc ion
and Falsi y- membe ship unc ions o e e y edge
)
,
( u
T
< min (
T
(u),
T
( ))
),
(
u
I
< min (
I
(u),
I
( ))
),( u
F
< max (
F
(u),
F
( )) and 0
),( u
T
+
)
,(
u
I
+
),( u
F
3
Figu e 1:Bijec i e Neu osophic G aph
De ini ion :2.3
The s eng h o he pa h wi h n edges is de ined as S(P) = (S(P1), S(P2), S(P3)) whe e
__________________________________________________________________________________________________
D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 287
S(P1) =
),
(
1 u
T
n
i
=
, S(P2) =
),(
1
u
I
n
i
=
, S(P3) =
),(
1
u
n
F
i
V
−
De ini ion :2.4
Le G be a Bijec i e neu osophic g aph. The connec ed be ween any wo e ices is de ined
by
)),(),,(),,((),( u u u u FIT
=
whe e
),( u
T
= Max(S(P1)),
)
,
( u
I
= Max(S(P2)),
),
(
u
F
= Min(S(P3))
De ini ion:2.5
Le G be a bijec i e neu osophic g aph. An edge o G is said o T-b idge
i
),( u
T
<
),(
' u
T
whe e
)
,(
' u
T
is he connec edness be ween u and by emo ing any edge.
An edge o G is said o I-b idge
i
),(
u
I
<
),(
' u
I
whe e
),(
' u
I
is he connec edness be ween u and by emo ing any edge.
An edge o G is said o F-b idge
i
),
(
u
F
>
),(
' u
F
whe e
),(
' u
F
is he connec edness be ween u and by emo ing any edge.
De ini ion: 2.6
An edge o G is said o be a neu osophic b idge i i is T- b idge, I-b idge and F- b idge.
3. An Algo i hm o inding he b idges o any bijec i e neu osophic g aph
Inpu : A bijec i e neu osophic g aph G = (V, E, T, I, F), whe e each edge has associa ed
u h-membe ship T, inde e minacy-membe ship I, and alsi y-membe ship F.
Ou pu : Classi ica ion o each b idge as T-B idge, I-B idge, o F-B idge.
S ep 1: Begin wi h a bijec i e neu osophic g aph G. Selec an a bi a y c isp cycle C*
consis ing o n- edges, whe e n ≥ 3
S ep 2: I T- b idge o I- b idge o G a e equi ed, hen iden i y an edge
T
=
T
n
i
1=
o
I
=
I
n
i
1=
by conside ing all he edges o C* espec i ely.
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D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 288
S ep 3: Remo e he iden i ied edge η om G.
S ep 4: Choose ano he cycle C* in G wi h any numbe o edges and epea s ep 2 & s ep 3
un il no such cycle emains in G.
S ep 5: A e emo al o all η’s om G, he esul ing g aph comp ises he T- b idges o I –
b idges o G espec i ely.
S ep 6: I F- b idges is equi ed o he chosen g aph hen ind
F
=
F
i
n
V
1
−
by conside ing all he edges o C*.
S ep 7: Repea s ep 3 and s ep 4.
S ep 8: Upon emo al o all
F
’s om he bijec i e neu osophic g aph G, he esul ing
g aph comp ises he F- b idges o G.
The T- b idges and I- b idges a e he s onges connec ions be ween he e ices in he g aph
G and F – b idges is he weakes connec ions be ween he e ices in he g aph G.
The esul ing g aph which is ob ained is he T- maximum spanning sub g aph o I- maximum
spanning sub g aph o G. Also one can ob ain he F-minimum spanning subg aph o G.
Example:3.1
Figu e 2:Bijec i e neu osophic g aph
Le us ind he T- b idges o he abo e igu e 2 by Conside ing he cycle C1* = 1,, 3, 4, 5, 1
o leng h 4.
He e η1= min(0.13, 0.15, 0.80, 0.59)
η1= 0.13
__________________________________________________________________________________________________
D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 289
Figu e 3
In he abo e igu e 3 edge e7 wi h 0.13 u h membe ship alue has o be emo ed om G.
Le he nex cycle be C2* = 1, 4, 5, 1 o leng h 3.
He e η2= min(0.55, 0.80, 0.59)
η2= 0.55
Figu e 4
Clea ly in he abo e igu e 4 he edge e4 wi h 0.55 u h membe ship alue has o be emo ed
om G. So Le he nex cycle be C3* = 1, 2, 3, 4, 5, 1 o leng h 5.
He e η3= min(0.58, 0.33, 0.15, 0.80, 0.59)
η3= 0.15
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D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 290
Figu e 5
Clea ly in he abo e igu e 5 he edge e3 wi h 0.15 as u h membe ship alue has o be
emo ed om G. Since no cycles emains a e he emo al o edges, he ollowing esul ing
g aph ep esen s all he T-B idges o G
Figu e 6: T-B idges o G
Hence he T- B idges o G a e ( 1, 2) , ( 2, 3) , ( 4, 5) and ( 5, 1)
I he abo e algo i hm is applied o Inde e minancy hen he I- B idge o G can be ob ained.
Hence he I- B idges o G a e ( 1, 3) , ( 1, 5) , ( 2, 3) and ( 3, 4)
Simila ly F-B idges o G a e ( 1, 5) , ( 1, 4) , ( 1, 2) and ( 2, 3).
In some p ac ical si ua ion i i is necessa y o conside all he membe ship unc ions in he
same ime hen we can use sco e unc ion o ind he b idges o bijec i e neu osophic g aph.
De ini ion:[12]
The sco e unc ion is de ined as
S(u) =
3
)()()( uuu FIT
++
The same algo i hm can be used o ind he b idges o he bijec i e neu osophic g aph a e
ind he sco e unc ion o all he e ices and edges.
__________________________________________________________________________________________________
D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 291
Example: Now le us ind he b idges o he g aph gi en in igu e a e inding he sco e
unc ion
S( 1) =
3
)()()( 111 FIT
++
=
3
30.071.060.0 ++
= 0.54
S( 2) = 0.56, S( 3) = 0.58 and so on.
Figu e 7
Fo inding he b idges o G le us i s conside a cycle C1* = 1, 2, 3, 4, 5, 1 o leng h 5.
Figu e 8
Clea ly he edge e2 is o be emo ed as i has minimum alue among he o he edge alues
in he conside ed cycle. Now le us choose ano he cycle C2* = 1, 3, 4, 5, 1 o leng h 4.
__________________________________________________________________________________________________
D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
Neu osophic Se s and Sys ems, Vol. 96, 2026 292
Figu e 9
The abo e ma ked edge e3 is he nex edge wi h minimum sco e alue which has o be
emo ed nex . Now conside ano he cycle C3* = 1, 4, 5, 1 o leng h 3.
Figu e 10
Clea ly i he abo e ma ked edge is emo ed, hen he b idges o G a e ( 1, 2) , ( 1, 3) ,
( 4, 5) and ( 5, 1).
No e: The sco e unc ion is one o he me hods o de neu osophica ion. So sco e unc ion
need no be bijec i e.
4. Applica ions
Now, Le ’s use he abo e discussed algo i hm o ind a be e diagnosis o cance . Le ’s
conside a speci ic case in which we ha e a pa ien wi h he mo e o less equal symp oms o
In lamma ion o Gas oin es inal T ac (GT), Ch onic Cough(C) and Fa igue(F).i.e. As pe
he da a, he exhibi s 48% o he ypical symp oms associa ed wi h In lamma ion o
Gas oin es inal T ac , 40% clinical symp oms o ch onic cough and 42% o Fa igue.
And we suspec ha he pa ien migh be su e ing om Colo ec al Cance (CRC), Lung
Cance (LC), Tube culosis(TB) and In lamma o y Bowel Disease(IBD).i.e. The e is
app oxima ely equal chance o him o su e om all he abo e men ioned disease. In
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D.Rajalaxmi, V.Vijaya and S.Re a hi, An Algo i hmic App oach o B idge De ec ion in Bijec i e Neu osophic
G aph wi h Applica ion in Cance Diagnosis
