Graph theory may help to address a wide range of real-world issues in computer applications, systems analysis, computer networks, transportation, operations research and economics. A graph is essentially a relational model that is used to describe real-world difficulties requiring relationships between entities. The graph’s vertices and edges represent the items and their relationships, respectively. The information available in many optimization situations is inaccurate or inexact for a number of reasons, including information loss, a lack of supporting evidence, flawed statistical data, and inadequate information. In general, knowledge about a real-life scenario may be ambiguous.

Cantor proposed the fundamental concept of classical graph theory. In a classical graph, each vertex or edge can have one of two outcomes: it is either part of the graph or not. Consequently, standard graphs cannot describe uncertain optimization situations. Real-world problems are typically unexpected, making it difficult to mimic them using conventional graphs. Zadeh [23] proposed fuzzy set theory. According to Smarandache’s theory, neutrosophic sets are a useful mathematical tool for dealing with confusing, inconsistent, and incomplete data in the real world. Neutrosophic sets in the real standard or nonstandard unit interval [0, 1] are defined separately by a truth-membership function (t), indeterminacy-membership function (i) and falsity-membership function (f). Wang created the concept of single-valued neutrosophic sets, a subclass of neutrosophic sets, to help with the use of NS in practical applications. Satham Hussain [15] introduced the concepts of neutrosophic vague graphs and strong neutrosophic vague graphs to help motivate the prior study.

Nagoor Gani [9, 10] defines regular and irregular fuzzy graphs and discusses the features of neighborly regular and irregular fuzzy graphs. Then, Liangsong Huang [7] introduced a research of regular and irregular neutrosophic graphs, the two types of degree, dm-regular, tdm-regular and m-highly irregular neutrosophic graphs, and certain features are presented.

Suriya [21] defines complex neutrosophic vague soft graphs, strong complex neutrosophic vague soft graphs, and constant complex neutrosophic vague soft graphs. Furthermore, after careful examination, we discovered some remarkable characteristics of the complement and self-complement of the complex neutrosophic vague soft graph.

Yahya [22] the Randic index of vague graphs and subgraphs are introduced along with their features. This study examines the top and lower bounds of the Randic index of vague graphs using isomorphic features. Randic indices for directed ambiguous graphs are introduced. Finally, an application of Randic index in building was given.

Vague graphs and neutrosophic graphs are sophisticated mathematical constructs that describe uncertainty in real-world networks. A vague graph is based on vague set theory, with each vertex or edge represented by a pair of truth-membership and false-membership degrees, whose sum cannot exceed one. The remaining implicitly denotes doubt or reluctance. Neutosophic graphs, on the other hand, take this notion a step further by incorporating three independent components: truth (T), indeterminacy (I) and falsity (F), with no restrictions on their total. This additional degree of flexibility enables neutrosophic graphs to better describe incomplete, inconsistent or contradictory information than vague graphs. While vague graphs are appropriate for moderate levels of uncertainty, where hesitation can be estimated using known truth and falsity values, neutrosophic graphs provide a richer and more flexible framework for highly uncertain or indeterminate environments, making them especially useful in fields such as artificial intelligence, decision-making systems and medical diagnosis.

We present the Randic index and neutrosophic subgraph features. This study examines a truth-membership function, indeterminacy-membership function and falsity-membership function of the Randic index in neutrosophic graphs using various isomorphic features. Randic indexes for neutrosophic graphs and neutrosophic digraphs are introduced then theorems as followed by example also explend. An application of Randic index in building is shown.

This section provides a basic description and example to support the key findings.

Definition 1. Where PA : X → [0, 1] and RA : X → [0, 1] are true membership and falsehood membership functions, respectively, such that a vague set A on a non-empty set X is a pair (PA, RA).

0 ≤ PA(x) + RA(x) ≤ 1

for every x ∈ X.

Consider two non-empty sets, X and Y . The criterion that a vague relation B of X to Y meets is that R = (PB, RB), where PB : X × Y → [0, 1] and RB : X × Y → [0, 1] is a vague set B on X × Y

0 ≤ PB(x, y) + RB(x, y) ≤ 1 for any x, y ∈ X.

Definition 2. Consider the graph G = (V, E). When J = (PJ , RJ ) is a vague set on V and K = (PK, RK) is a vague set on E ⊆ V × V, such that for each xy ∈ E,

PK(xy) ≤ min{PJ (x), PJ (y)} and RK(xy) ≤ min{RJ (x), RJ (y)}.

Definition 3. A neutrosophic set A contained neutrosophic set B, i.e., A ⊆ B if for all x ∈ X, PA(x) ≤ PB(x), QA(x) ≥ QB(x) and RA(x) ≥ RB(x).

Definition 4. Let X be a space of point (objects) and let X represent the generic elements in X. There is only one neutrosophic set A in X and its truth membership function defines it as the PA(x), a membership function for indeterminacy QA(x) and the membership function for falsehood RA(x).

For each point x in X, PA(x), QA(x), RA(x) ∈ [0, 1] A = {x, PA(x), QA(x), RA(x)} and 0 ≤ PA(x) + QA(x) + RA(x) ≤ 3.

Definition 5. The Randic index of a neutrosophic graph G = (S, K) is shown by RI(G) and explained as follows:

RI(G) = (RIP(G), RIQ(G), RIR(G))

where, RIP(G) = PS(ci)PS(cj)DP(ci)DP(cj)

i̸=j,cicj ∈E

RIR(G) = RS(ci)RS(cj)DR(ci)DR(cj)

i̸=j,cicj ∈E

where true and false parts of the node ci degrees are DP and DR, respectively.

Definition 6. The Randic index of a neutrosophic graph G = (S, K) is shown by RI(G) and explained as follows:

RI(G) = (RIP(G), RIQ(G), RIR(G))

where, RIP(G) = p (PS(ci)PS(cj)DP(ci)DP(cj))

RIQ(G) = p (QS(ci)QS(cj)DQ(ci)DQ(cj))

i̸=j,cicj ∈E

RIR(G) = p (RS(ci)RS(cj)DR(ci)DR(cj))

i̸=j,cicj ∈E

that the true, indeterminacy and false parts of the node ci degrees are DP, DQ and DR, respectively.

Example 1. Assume the neutrosophic graphs G as figure 1. Here, D(v) = (0.4, 1.5, 1.4), D(w) = (0.5, 1.5, 1.5), D(x) = (0.5, 2.4, 2.4), D(y) = (0.4, 2.5, 2.3) and D(z) = (0.3, 1.8, 1.5).

p = 6.455 + 8.165 + 11.180 + 8.839 + 11.785 + 10.541 (PS(ci)PS(cj)DP(ci)DP(cj))

i̸=j,cicj ∈E

= 56.965

p = 1.260 + 0.813 + 0.657 + 1.017 + 0.831 + 0.813 (QS(ci)QS(cj)DQ(ci)DQ(cj))

i̸=j,cicj ∈E = 4.578

p = 1.260 + 0.813 + 0.657 + 1.017 + 0.831 + 0.813 (RS(ci)RS(cj)DR(ci)DR(cj))

i̸=j,cicj ∈E

= 5.397

RI(G) = (RIP(G), RIQ(G), RIR(G)) = (56.965, 4.578, 5.391).

′ ′ ′ ′

Theorem 1. Assume G = (S, K) be a connected neutrosophic Graph and G = (S , K ) such that V = V −{xn)}, xn ∈

′ ′ ′

V with |V | = n. Thus, RIP(G) ≥ RIP(G ), RIQ(G) ≥ RIQ(G ) and RIR(G) ≥ RIR(G ).

Proof: Since G contains n notes i.e.,V = {x1, x2, ..., xn}, V = {x1, x2, ..., xn−1}. Hence V must be a subset of V as so G is a neutrosophic subgraph of G. Hence,

PS(xi) = PS′ (xi), QS(xi) = QS′ (xi), RS(xi) = RS′ (xi)

PK(xixj) = PK′ (xixj), QK(xixj) = QK′ (xixj), RK(xixj) = RK′ (xixj) for all xi ∈ V and for all xixj ∈ E .

X X X DP(xi) = PK(xixj), DQ(xi) = QK(xixj), DR(xi) = RK(xixj)

xixj ∈E xixj ∈E

i.e. DP(xi), DQ(xi), DR(xi) are sume of truth, indetetminacy and falsity membership values for edges occurring in xi in G, respectively. Hence, DR(xi)DR(xj) is positive real number. Similarly, we can show that D (xi)D (xj) is also positive real number. Thus,

p ≥ 0, q ≥ 0, (PS(ci)PS(cj)DP(ci)DP(cj)) ′ ′ (PS′ (ci)PS′ (cj)D (ci)D (cj))

p ≥ 0, p ≥ 0, (Q (c )Q (c )D (c )D (c )) (QS′ (ci)QS′ (cj)D (ci)D (cj))

S i S j Q i Q j Q Q

p ≥ 0, p ≥ 0.

(R (c )R (c )D (c )D (c )) (RS′ (ci)RS′ (cj)D (ci)D (cj))

S i S j R i R j R R So, X 1 X 1 p ≥ q (PS(ci)PS(cj)DP(ci)DP(cj)) ′ ′ (PS′ (ci)PS′ (cj)D (ci)D (cj))

1≤i̸=j≥a,cicj ∈E 1≤i̸=j≥a,cicj ∈E P P

X 1 X 1 p ≥ p (Q (c )Q (c )D (c )D (c )) (QS′ (ci)QS′ (cj)D (ci)D (cj))

1≤i̸=j≥a,c c ∈E S i S j Q i Q j 1≤i̸=j≥a,c c ∈E Q Q

i j i j

X 1 X 1 p ≥ p (R (c )R (c )D (c )D (c )) (RS′ (ci)RS′ (cj)D (ci)D (cj))

1≤i̸=j≥a,c c ∈E S i S j R i R j 1≤i̸=j≥a,c c ∈E R R

i j i j

′ ′ ′

Therefore, RIP(G) ≥ RIP(G ), RIQ(G) ≥ RIQ(G ) and RIR(G) ≥ RIR(G ).

′ ′ ′

Example 2. Assume the neutrosophic subgraph G = (S , K ) of figure 2 and neutrosophic graph G of figure 1. Clearly,

PS(xi) ≥ PS′ (xi), QS(xi) ≤ QS′ (xi), RS(xi) ≤ RS′ (xi), ∀xi ∈ V

PK(xixj) ≥ PK′ (xixj), QK(xixj) ≤ QK′ (xixj), RK(xixj) ≤ RK′ (xixj), ∀xixj ∈ E .

Hence, G is a neutrosophic subgraph of neutrosophic graph G.

RIP(G ) = 6.455 + 10.541 + 20.412 + 12.500 = 49.908 RIQ(G ) = 1.029 + 0.932 + 0.988 + 1.091 = 4.04 RIR(G ) = 1.260 + 0.966 + 0.936 + 1.220 = 4.382

′ ′ ′

Therefore, RIP(G) ≥ RIP(G ), RIQ(G) ≥ RIQ(G ) and RIR(G) ≥ RIR(G ).

Theorem 2. Assume G = (S, K) is a neutrosophic graphs of the graph G∗ = (V,E). Then, RIP(G) ≥ ,

(a−1)

RIQ(G) ≥ and RIR(G) ≥ , where, a = |V | and b = |E|.

(a−1) (a−1)

Proof: Since G is a neutrosophic graph, 0 ≤ PS(bi) ≤ 1, 0 ≤ PS(bj) ≤ 1, 0 ≤ QS(bi) ≤ 1, 0 ≤ QS(bj) ≤ 1, 0 ≤ RS(bi) ≤ 1, 0 ≤ RS(bj) ≤ 1 ∀ bi, bj ∈ E . So,

PS(bi)PS(bj) ≤ 1, QS(bi)QS(bj) ≤ 1, RS(bi)RS(bj) ≤ 1 (i)

Since |V | = a, each vertex is connected to at most (a − 1) vertices. But, DP(bi) = P(b b ), D (b )) =

i j Q i bibj ∈E

Q(bibj), DR(bi)) = R(bibj), so DP(bi) ≤ a − 1, DQ(bi) ≤ a − 1 and DR(bi) ≤ a − 1, ∀ bi ∈ V . This

bibj ∈E bibj ∈E

implies that

2 2 2

DP(bi)DP(bj) ≤ (a − 1) , DQ(bi)DQ(bj) ≤ (a − 1) , DR(bi)DR(bj) ≤ (a − 1) (ii) So, from (i) and (ii), we get:

PS(bi)PS(bj)DP(bi)DP(bj) ≤ (a − 1) ,

QS(bi)QS(bj)DQ(bi)DQ(bj) ≤ (a − 1) ,

RS(bi)RS(bj)DR(bi)DR(bj) ≤ (a − 1) .

Hence,

p ≥ , PS(bi)PS(bj)DP(bi)DP(bj) (a − 1)

p ≥ , QS(bi)QS(bj)DQ(bi)DQ(bj) (a − 1)

p ≥ .

RS(bi)RS(bj)DR(bi)DR(bj) (a − 1)

Therefore, X 1 b p ≥ , PS(bi)PS(bj)DP(bi)DP(bj) (a − 1)

X 1 b p ≥ , QS(bi)QS(bj)DQ(bi)DQ(bj) (a − 1)

X 1 b p ≥ .

RS(bi)RS(bj)DR(bi)DR(bj) (a − 1)

b b b

Therefore, RIP(G) ≥ , RIQ(G) ≥ and RIR(G) ≥ .

(a−1) (a−1) (a−1)

Example 3. Assume the neutrosophic graph G in Example 1. Here, a = |V | = 5, b = |E| = 6, RIP(G) = 56.965, RIQ(G) = 4.578, RIR(G) = 5.391 and = = 1.2.

(a−1) 5 b b b

Therefore, RIP(G) > , RIQ(G) > and RIR(G) > .

(a−1) (a−1) (a−1)

Theorem 3. Assume G = (S, K) be a strong neutrosophic graph so that S is a constant function and |V | = a. Then,

a a a

RIP(G) ≤ 2 , RIQ(G) ≤ 2 and RIR(G) ≤ 2 , where, (v1, v2, v3) = (PS(bi), QS(bi), RS(bi)), bi ∈ V.

2v 2v 2v

1 2 3

Proof: Since a strong neutrosophic graph is generally a neutrosophic subgraph of the corresponding complete neutrosophic

a a a

graph, using theorem 2, it is simple to prove that RIP(G) ≤ 2 , RIQ(G) ≤ 2 and RIR(G) ≤ 2 .

2v 2v 2v

1 2 3

a a a

Theorem 4. Assume G be a complete neutrosophic graph so that S is a constant function. Then RI(G) = 2 , 2 , 2 ,

2v 2v 2v

1 2 3

where a = |V | and (v1, v2, v3) = (PS(bi), QS(bi), RS(bi)), bi ∈ V.

Proof: Since S is a constant and PS(bi) = v1, QS(bi) = v2, RS(bi) = v3, bi ∈ V. Since G is complete neutrosophic graph, PK(bibj) = {PS(bi) ∧ PS(bj)}, QK(bibj) = {QS(bi) ∧ QS(bj)} and RK(bibj) = {RS(bi) ∧ RS(bj)}, ∀bibj ∈ E.

a(a−1) a(a−1)

Since G is complete and |V | = a, there are pairs of vertices and edges. Additionally, each vertices in G is adjacent to (a − 1) vertices. Then,

DP(bi) = PK(bibj) = v1 · v1 · · · v1(a − 1 times) = (a − 1)v1

bibj ∈E

DQ(bi) = QK(bibj) = v2 · v2 · · · v2(a − 1 times) = (a − 1)v2

bibj ∈E

DR(bi) = RK(bibj) = v3 · v3 · · · v3(a − 1 times) = (a − 1)v3

bibj ∈E Therefore,

RIP(G) = p PS(bi)PS(bj)DP(bi)DP(bj)

a(a − 1) 1 = · p 2 (v v (a − 1)v (a − 1)v )

1 1 1 1

RIQ(G) = p QS(bi)QS(bj)DQ(bi)DQ(bj)

a(a − 1) 1 = · p 2 (v v (a − 1)v (a − 1)v )

2 2 2 2

RIR(G) = p RS(bi)RS(bj)DR(bi)DR(bj)

a(a − 1) 1 = · p 2 (v v (a − 1)v (a − 1)v )

3 3 3 3

a a a

Hence, RI(G) = 2 , 2 , 2 .

2v 2v 2v

1 2 3

Proposition 1. If two neutrosophic graphs G and G are isomorphic to each other, then, RI(G) = RI(G )

′ ′ ′

Proof: Assume G = (S, K) and G = (S , K ) be two isomorphic neutrosophic graphs. Then , there is a bijection l : G → G so that PS(bi) = PS′ (l(bi)) = PS(b ),

QS(bi) = QS′ (l(bi)) = QS(b ),

RS(bi) = RS′ (l(bi)) = RS(b ), ∀ bi ∈ V and PK(bibj) = PK′ (l(bi)l(bj)) = PK(b b ),

QK(bibj) = QK′ (l(bi)l(bj)) = QK(b b ),

RK(bibj) = RK′ (l(bi)l(bj)) = RK(b b ), ∀ bibj ∈ E.

′ ′ ′

So, for each bi ∈ V there exists a vertex bi ∈ V so that D(bi) = D(bi).

Thus, we have the proof.

Example 4. Assume the neutrosophic graphs G and G of figure 3. Here, PS′ (l(bi)) = PS(ai), QS′ (l(bi)) = QS(ai), RS′ (l(bi)) = RS(ai), PK′ (l(bi)l(bj)) = PK(aiaj), QK′ (l(bi)l(bj)) = QK(aiaj), RK′ (l(bi)l(bj)) = RK(aiaj), ∀1 ≥ i, j ≤ 3. Here, RIP(G) = 25.000 + 35.355 + 35.355 = 95.710 = RIP(G )

RIQ(G) = 1.207 + 1.304 + 1.102 = 3.613 = RIQ(G )

RIR(G) = 1.181 + 1.056 + 1.398 = 3.635 = RIR(G ) Thus, RI(G) = RI(G ).

Theorem 5. Assume G = (S, K) be a neutrosophic so that

µ1 = min{DP(bi)}, µ2 = min{DQ(bi)}, µ3 = min{DR(bi)}

η1 = max{DP(bi)}, η2 = max{DQ(bi)}, η3 = max{DR(bi)}

∀bi ∈ V.

f1 f1 f2 f2 f1 f3

Then, ≤ RIP(G) ≤ , ≤ RIQ(G) ≤ and ≤ RIR(G) ≤ , where

η1 µ1 η2 µ2 η3 µ3

X 1 X 1 f1 = p , f2 = p , PS(bi)PS(bj) QS(bi)QS(bj)

i̸=j,bibj ∈E i̸=j,bibj ∈E

f3 = p .

RS(bi)RS(bj)

Proof: Since G is a neutrosophic graph and bi ∈ V , X X X DP(bi) = PK(bibj), DQ(bi) = QK(bibj), DR(bi) = RK(bibj).

bibj ∈E bibj ∈E bibj ∈E

Again, µ1 = min{DP(bi)}, µ2 = min{DQ(bi)}, µ3 = min{DR(bi)}

η1 = max{DP(bi)}, η2 = max{DQ(bi)}, η3 = max{DR(bi)}

∀bi ∈ V , so we have µ1 ≤ DP(bi) ≤ η1, µ2 ≤ DQ(bi) ≤ η2, µ3 ≤ DR(bi) ≤ η3,

∀bi ∈ V . Then

2 2 2 2 2 2

µ1 ≤ DP(bi)DP(bj) ≤ η1, µ2 ≤ DQ(bi)DQ(bj) ≤ η2, µ3 ≤ DR(bi)DR(bj) ≤ η3,

∀bi, bj ∈ V . Thus, P(bi)P(bj)µ1 ≤ P(bi)P(bj)DP(bi)DP(bj) ≤ P(bi)P(bj)η1,

Q(bi)Q(bj)µ2 ≤ Q(bi)Q(bj)DQ(bi)DQ(bj) ≤ Q(bi)Q(bj)η2,

R(bi)R(bj)µ3 ≤ R(bi)R(bj)DR(bi)DR(bj) ≤ R(bi)R(bj)η3,

∀bi, bj ∈ V . So, X 1 X 1 p ≤ p PS(bi)PS(bj)η PS(bi)PS(bj)DP(bi)DP(bj)

i̸=j,b b ∈E 1 i̸=j,b b ∈E

i j i j

PS(bi)PS(bj)µ1 X 1 X 1 p ≤ p QS(bi)QS(bj)η QS(bi)QS(bj)DQ(bi)DQ(bj)

i̸=j,b b ∈E 2 i̸=j,b b ∈E

i j i j

QS(bi)QS(bj)µ2

X 1 X 1 p ≤ p RS(bi)RS(bj)η RS(bi)RS(bj)DR(bi)DR(bj)

i̸=j,b b ∈E 3 i̸=j,b b ∈E

i j i j

RS(bi)RS(bj)µ3

Hence, X 1 1 X 1 1 p · ≤ RIP(G) ≤ p · PS(bi)PS(bj) η1 PS(bi)PS(bj) µ1

i̸=j,bibj ∈E i̸=j,bibj ∈E

X 1 1 X 1 1 p · ≤ RIQ(G) ≤ p · QS(bi)QS(bj) η2 QS(bi)QS(bj) µ2

i̸=j,bibj ∈E i̸=j,bibj ∈E

X 1 1 X 1 1 p · ≤ RIR(G) ≤ p · RS(bi)RS(bj) η3 RS(bi)RS(bj) µ3

i̸=j,bibj ∈E i̸=j,bibj ∈E f1 f1 f2 f2 f1 f3

Therefore, ≤ RIP(G) ≤ , ≤ RIQ(G) ≤ and ≤ RIR(G) ≤ .

η1 µ1 η2 µ2 η3 µ3

Example 5. Assume the neutrosophic graph G of Example 4. Here, V = {b1, b2, b3}, E = {b1b2, b2b3, b1b3}

DP(b1), DQ(b1), DR(b1) = DP(b2), DQ(b2), DR(b2) = DP(b3), DQ(b3), DR(b3) = (0.2, 1.4, 1.6)

So, µ1 = η1 = 0.2, µ2 = η2 = 1.4, µ3 = η3 = 1.6 RIP(G) = 95.710, RIQ(G) = 3.613, RIR(G) = 3.635

f1 = p PS(bi)PS(bj)

1 1 1 = √ + √ + √ 0.2 × 0.2 0.2 × 0.1 0.1 × 0.2 = 19.142

f1 = p QS(bi)QS(bj)

1 1 1 = √ + √ + √ 0.7 × 0.5 0.5 × 0.6 0.6 × 0.7 = 5.059

f1 = p RS(bi)RS(bj)

1 1 1 = √ + √ + √ 0.4 × 0.7 0.7 × 0.5 0.5 × 0.4 = 5.816

f1 f1 f2 f2 f3 f3

So, = = 95.71, = = 3.613, = = 3.635.

µ1 η1 µ2 η2 µ3 η3

Therefore, f1 f1 RIP(G) = = = 95.71 µ1 η1 f2 f2 RIQ(G) = = = 3.613 µ2 η2

f3 f3 RIR(G) = = = 3.635.

µ3 η3

Theorem 6. Assume G = (S, K) be a neutrosophic graph so that |V | = a. Then,

1 X 1 RIP(G) ≥ a − 1 PS(bi)PS(bj)

bibj ∈E

1 X 1 RIQ(G) ≥ a − 1 QS(bi)QS(bj)

bibj ∈E

1 X 1 RIR(G) ≥ .

a − 1 RS(bi)RS(bj)

bibj ∈E

Proof: Since |V | = a, consider, every vertex of G is connected with at most a − 1 other vertices. Therefore,

PK(bibj) ≤ (a − 1)PS(bi)

bibj ∈E

QK(bibj) ≤ (a − 1)QS(bi)

bibj ∈E

RK(bibj) ≤ (a − 1)RS(bi).

bibj ∈E

So, DP(bi) ≤ (a − 1)PS(bi), DQ(bi) ≤ (a − 1)QS(bi), DR(bi) ≤ (a − 1)RS(bi)

∀bi ∈ V. Hence,

PS(bi)PS(bj)DP(bi)DP(bj) ≤ PS(bi)PS(bj)(a − 1)PS(bi)(a − 1)PS(bj)

QS(bi)QS(bj)DQ(bi)DP(bj) ≤ QS(bi)QS(bj)(a − 1)QS(bi)(a − 1)QS(bj)

RS(bi)RS(bj)DR(bi)DR(bj) ≤ RS(bi)RS(bj)(a − 1)RS(bi)(a − 1)RS(bj)

∀bibj ∈ E. So,

X 1 1 X 1

(PS(bi)PSbj)DP(bi)DP(bj)) a − 1 PS(bi)PS(bj)

i̸=j,bibj ∈E bibj ∈E

X 1 1 X 1

(QS(bi)QSbj)DQ(bi)DQ(bj)) a − 1 QS(bi)QS(bj)

i̸=j,bibj ∈E bibj ∈E

X 1 1 X 1

(RS(bi)RSbj)DR(bi)DR(bj)) a − 1 RS(bi)RS(bj)

i̸=j,bibj ∈E bibj ∈E

Thus, 1 X 1 RIP(G) ≥ a − 1 PS(bi)PS(bj)

bibj ∈E

1 X 1 RIQ(G) ≥ a − 1 QS(bi)QS(bj)

bibj ∈E

1 X 1 RIR(G) ≥ .

a − 1 RS(bi)RS(bj)

bibj ∈E −→ −→ Definition 7. The degree of a vertex bi in a neutrosophic digraph G = (S, K) is D(bi) = (DP(bi), DQ(bi), DR(bi)), where X −−→ −−→ DP(bi) = (PK(bibj) + PK(bjbi))

i̸=j

X −−→ −−→ DQ(bi) = (QK(bibj) + QK(bjbi))

i̸=j

X −−→ −−→ DR(bi) = (RK(bibj) + RK(bjbi)).

i̸=j

Example 6. Assume the neutrosophic digraph G of figure 4. Here, PS(bi), QS(bi), RS(bi) , i = 1, 2, 3.

DP(b1) = DP(b2) = DP(b3) = 0.6

DQ(b1) = DQ(b2) = DQ(b3) = 3.0

DR(b1) = DR(b2) = DR(b3) = 2.6

D(b1) = D(b2) = D(b3) = (0.6, 3.0, 2.6).

→− →− →− Definition 8. The Randic index of a neutrosophic digraph G = (S, K) is denoted by RI(G) and is denoted as:

→− →− →− →− RI(G) = (RIP(G), RIQ(G), RIR(G))

 √ ,  (PS (bi)PS bj )DP(bi)DP(bj )) i̸=j,bibj ∈E   P

= (Q (b )Q b )D (b )D (b ))

S i S j Q i Q j

i̸=j,bibj ∈E   √ .

(RS (bi)RS bj )DR(bi)DR(bj ))

Example 7. Consider the neutrosophic digraph G of example 6. We have:

−→ X 1 RIP(G) = p (PS(bi)PSbj)DP(bi)DP(bj))

= 3 · p (0.2 × 0.2 × 0.6 × 0.6) = 25 −→ X 1 RIQ(G) = p (QS(bi)QSbj)DQ(bi)DQ(bj))

= 3 · p (0.7 × 0.7 × 3.0 × 3.0) = 1.429

−→ X 1 RIR(G) = p (RS(bi)RSbj)DR(bi)DR(bj))

= 3 · p (0.5 × 0.5 × 2.6 × 2.6) = 2.308

−→ −→ −→ −→ Hence, RI(G) = (RIP(G), RIQ(G), RIR(G)) = (25, 1.429, 2.308).

−→ −→ Theorem 7. Assume G = (S, K) be a neutrosophic digraph with |V | = b so that S = (PS, QS, RS, ) is constant.

1 1 1

If D = (x1, x2, x3), ∀bi ∈ V , then RI(G) = b( , , ), where (v1, v2, v3) = (PS(bi), QS(bi), RS(bi)),

v1x1 v2x2 v3x3

∀bi ∈ V .

−→ −→ Proof: Let the neutrosophic digraph G = (S, K) so that S = (PS, QS, RS, ) is constant, D = (x1, x2, x3) and (v1, v2, v3) = (PS(bi), QS(bi), RS(bi)), ∀bi ∈ V .

−→ X 1 RIP(G) = p (PS(bi)PSbj)DP(bi)DP(bj))

v1v1x1x1

v1x1

−→ X 1 RIQ(G) = p (QS(bi)QSbj)DQ(bi)DQ(bj))

v2v2x2x2

v2x2

−→ X 1 RIR(G) = p (RS(bi)RSbj)DR(bi)DR(bj))

v3v3x3x3

v3x3 −→ −→ −→ −→

1 1 1

Therefore, RI(G) = (RIP(G), RIQ(G), RIR(G)) = b , , .

v1x1 v2x2 v3x3 Example 8. Let the neutrosophic digraph G of example 6. Here, |V | = b = 3, degree of every vertex =(x1, x2, x3)= (0.6,3.0,2.6), membership value of every vertex =(v1, v2, v3) = (0.2, 0.7, 0.5) and RI(G) = (25.000, 1.129, 2.308). Now,

= = 25.000 v1x1 0.2 × 0.6

= = 1.429 v2x2 0.7 × 3.0

= = 2.308 v3x3 0.5 × 2.6 Therefore, theorem 6 is verified.

Theorem 8. Assume G = (S, K) be a regular and totally regular neutrosophic graph so that there is an edge between

a(a−1) 1 1 1

each pair of vertices and |V | = a. Then, RI(G) = , , , where x1 = DP(bi), x2 = DQ(bi),

2 v1x1 v2x2 v3x3

x3 = DR(bi), v1 = P(bi), v2 = Q(bi), v3 = R(bi), ∀bi ∈ V .

Proof: Since G is regular, DP(bi) = x1, DQ(bi) = x2, DR(bi) = x3, ∀bi ∈ V . Again, G is totally regular neutrosophic graph, so let tDP(bi) = h1, tDQ(bi) = h2, tDR(bi) = h3, ∀bi ∈ V . We no that, tDP(bi) = DP(bi) + PS(bi), tDQ(bi) = DQ(bi) + QS(bi), tDR(bi) = DR(bi) + RS(bi), ∀bi ∈ V .

Thus, v1 = PS(bi) = tDP(bi) − DP(bi) = h1 − x1

v2 = QS(bi) = tDQ(bi) − DQ(bi) = h2 − x2

v3 = RS(bi) = tDR(bi) − DR(bi) = h3 − x3 Hence, S is constant function and v1 = PS(bi), v2 = QS(bi), v3 = RS(bi), ∀bi ∈ V . Since |V | = a and there is an edge

a(a−1)

between each pair of vertices in G, so there are number of edges in G. Now

−→ X 1 RIP(G) = p (PS(bi)PSbj)DP(bi)DP(bj))

a(a − 1) 1 = · √ 2 v1v1x1x1 a(a − 1) 2v1x1

−→ X 1 RIQ(G) = p (QS(bi)QSbj)DQ(bi)DQ(bj))

a(a − 1) 1 = · √ 2 v2v2x2x2 a(a − 1) 2v2x2

−→ X 1 RIR(G) = p (RS(bi)RSbj)DR(bi)DR(bj))

a(a − 1) 1 = · √ 2 v3v3x3x3 a(a − 1) 2v3x3

a(a−1) 1 1 1

Therefore, RI(G) = (RIP(G), RIQ(G), RIR(G)) = , , .

2 v1x1 v2x2 v3x3

A government agency intends to construct a waste processing plant in an area that includes several medium-sized cities. The ideal location should have public and administrative backing, be easily accessible to neighboring cities for efficient waste transportation, have existing infrastructure (partial facilities, land, staff) and minimize environmental and logistical concerns.

The example demonstrates how to utilize a neutrosophic graph and the Randic index to select the best city for building a waste management plant. The selection process takes into account a variety of parameters, including public support, infrastructure availability, road connection and environmental concerns, which are represented as nodes (cities) and edges (roads) with neutrosophic weights signifying a truth, indeterminacy and falsity-membership function. Cities such as C1, C2, C3, C4 and C5 are examined based on their population numbers and imprecise a truth, indeterminacy and falsity-membership ratings. The Randic index, a graph-theoretic metric, is used to determine node centrality and significance.

D(C1) = (0.2.1.6, 1.5),D(C2) = (0.6, 2.5, 2.4),D(C3) = (0.6, 2.4, 2.4),D(C4) = (0.5, 3.3, 3.4) and D(C5) = (0.3, 1.4, 1.7).

p = 14.434 + 22.361 + 4.811 + 6.455 + 7.454 + 9.623 + 12.910 (PS(Ci)PS(Cj)DP(Ci)DP(Cj))

i̸=j,CiCj ∈E

= 78.048

p = 0.747 + 0.736 + 0.589 + 0.465 + 0.548 + 0.996 + 0.786 (QS(Ci)QS(Cj)DQ(Ci)DQ(Cj))

i̸=j,CiCj ∈E

= 4.867

p = 0.962 + 0.639 + 0.704 + 0.554 + 0.468 + 0.936 + 0.735 (RS(Ci)RS(Cj)DR(Ci)DR(Cj))

i̸=j,CiCj ∈E

= 4.998

So,

RI(G) = (RIP(G), RIQ(G), RIR(G)) = (78.048, 4.867, 4.998) Then,

RIP(G − C1) = 78.048 − 12.434 − 22.361 = 41.250, RIP(G − C2) = 78.048 − 14.434 − 4.811 − 6.455 = 52.348, RIP(G − C3) = 78.048 − 4.811 − 7.454 − 9.623 = 56.160, RIP(G − C4) = 78.048 − 22.361 − 6.455 − 7.454 − 12.910 = 28.868 RIP(G − C5) = 78.048 − 9.623 − 12.910 = 55.515.

RIQ(G − C1) = 4.867 − 0.747 − 0.736 = 3.384, RIQ(G − C2) = 4.867 − 0.747 − 0.589 − 0.465 = 3.066, RIQ(G − C3) = 4.867 − 0.589 − 0.548 − 0.996 = 2.944, RIQ(G − C4) = 4.867 − 0.736 − 0.465 − 0.548 − 0.786 = 2.602, RIQ(G − C5) = 4.867 − 0.996 − 0.786 = 3.327.

RIR(G − C1) = 4.998 − 0.962 − 0.639 = 3.397, RIR(G − C2) = 4.998 − 0.962 − 0.704 − 0.554 = 2.778, RIR(G − C3) = 4.998 − 0.704 − 0.468 − 0.936 = 2.890, RIR(G − C4) = 4.998 − 0.639 − 0.554 − 0.468 − 0.735 = 2.602, RIR(G − C5) = 4.998 − 0.936 − 0.735 = 3.327.

The computation above shows that avoid nodes C1, C2, C3 and C5, this isn’t appropriate for constructing a waste processing plant. The most important and significant factor in preserving a high randix index is city C4, then the largest population and most transportation routes are found in city C4. the best location for the waste management facility is city C4.

Neutrosophic graphs are more precise, flexible, and compatible than fuzzy graphs. Neutrosophic graphs are commonly used in social networks to identify the most effective individuals within a group or organisation. The Connectivity Index has numerous uses in psychology, medical science, social groupings, and computer networks. This work presents randic indexes for neutrosophic graphs and subgraphs, along with their properties. a truth, indeterminacy and falsity-membership function of the randic index of vague graphs are investigated using some isomorphic properties. Additionally, the randic index of directed neutrosophic graphs is introduced. In the future, we plan to explore Eccentricity index and wiener index for neutrosophic incidence graphs.

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

Each author contribution equally to the piece. All authors have read and agreed to the published version of the manuscript.

This research was supported by the University Grants Commission (UGC) of India under the scheme CSIR-UGC JRF with reference number 231610081608. The first author gratefully acknowledges the financial support provided by UGC.

This manuscript has no associated data.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.