Preliminaries
Explanation 2.1:
BSVN sets

A BSVN set is explained as the membership functions represented as an object in W is denoted by <w,TP,IP,FP,TN,IN,FN>:wϵW the functions TP, IP, FPare mapping from W to [0, 1] and TN, IN, FNare mapping from W_{ } to [-1, 0].

Example. Let W=w1, w2, w3 and A=<w1,0.4,0.2,0.6,-0.03,-0.2,-0.01>,<w2, 0.6,0.4,0.2,-0.5,-0.3,-0.03>, _{ <w3,0.7,0.04,0.3,-0.8,-0.4,-0.05>is a BSVN set in W.}

Explanation 2.2: SVN relation on W:-

Let W be a non-empty set. Then we call mapping Z=( W, Tp, Ip, FP, TN, IN, FN)_{,} FNw:W X W→-1, 0 X [0, 1] is a BSVN relation on W such that

TZPw1,w2ϵ[0,1],IZPw1,w2ϵ[0,1],FZPw1,w2ϵ[0,1] and TZNw1,w2ϵ[-1,0],IZNw1,w2ϵ[-1,0],FZNw1,w2ϵ[-1,0]

Explanation 2.3: Let Z1=(Tz1P, Iz1P, Fz1P, Tz1N, Iz1N, Fz1N) and Z2=(Tz2P, Iz2P, Fz2P, Tz2N, Iz2N, Fz2N) be a BSVN graphs on a set W. If Z2 is a BSVN relation on Z1, then

TZ2P(w1, w2)≤min(TZ1Pw1, TZ1Pw2), IZ2P(w1, w2)≥max(IZ1Pw1, IZ1Pw2)_{,
} FZ2P(w1, w2)≥max(FZ1Pw1, FZ1Pw2) and TZ2N(w1, w2)≥max(TZ1Nw1, TZ1Nw2)_{,
} IZ2P(w1, w2)≤min(IZ1Nw1, IZ1Nw2)_{, } FZ2N(w1, w2)≤min(FZ1Nw1, FZ1Nw2)for all w1, w2∈W

Explanation 2.4: The symmetric property defined on BSVN relation Z on W is explained by

TZPw1, w2=TZP(w2, w1), IZPw1, w2=IZP(w2, w1), FZPw1, w2=FZP(w2, w1) and TZNw1, w2=TZN(w2, w1), IZNw1, w2=IZN(w2, w1), FZNw1, w2=FZN(w2, w1) for all w1, w2∈W

Explanation 2.5
: BSVN graph

The new graph in SVN is denoted by for all w1, w2∈W, G*=(V, E) is a pair G=(Z1, Z2), where Z1=(Tz1P, Iz1P, Fz1P, Tz1N, Iz1N, Fz1N) is a BSVNS in V and Z2=(Tz2P, Iz2P, Fz2P, Tz2N, Iz2N, Fz2N) is BSVNS in V2 defined as

TZ2P(w1, w2)≤min(TZ1Pw1, TZ1Pw2), IZ2P(w1, w2)≥max(IZ1Pw1, IZ1Pw2)_{,
} FZ2P(w1, w2)≥max(FZ1Pw1, FZ1Pw2) and TZ2N(w1, w2)≥max(TZ1Nw1, TZ1Nw2)_{,
} IZ2P(w1, w2)≤min(IZ1Nw1, IZ1Nw2)_{, } FZ2N(w1, w2)≤min(FZ1Nw1, FZ1Nw2)for all w1, w2∈W

The BSVNSG of an edge denoted by w1w2∈V2

Explanation 2.6 Let G=(Z1, Z2) be a BSVNSG and_{ } a, b∈V. A path P:a=w0, w1, w2, …,wk-1, wk=b in G is a sequence of distinct vertices such that TZ2P(wi-1, wi)>0, IZ2P(wi-1, wi)>0, FZ2P(wi-1, wi)>0, TZ2N(wi-1, wi)>0, IZ2N(wi-1, wi)>0, FZ2N(wi-1, wi)>0 where i=1, 2, 3, …, k and length of the path is k∈N(a positive integer)_{, }where a is called initial vertex and b is called terminal vertex in the path.

Explanation 2.7. A BSVN graph G=(Z1, Z2) of G*=(V, E) is called a strong BSVN graph if TZ2Pw1, w2=min(TZ1Pw1, TZ1Pw2), IZ2Pw1, w2=max(IZ1Pw1, IZ1Pw2)_{, } FZ2Pw1, w2=max(FZ1Pw1, FZ1Pw2) and TZ2Nw1, w2=max(TZ1Nw1, TZ1Nw2)_{,
} IZ2Pw1, w2=min(IZ1Nw1, IZ1Nw2)_{, } FZ2Nw1, w2=min(FZ1Nw1, FZ1Nw2)for all w1, w2∈W

If P:a=w0, w1, w2, …,wk-1, wk=b be a path of length k between a and b then (TZ2Pa, b,IZ2Pa, b,FZ2Pa, b)k and (TZ2Na, b,IZ2Na, b,FZ2Na, b)k is defined as

TZ2P(a,b),IZ2P(a,b),FZ2P(a,b)k=supTZ2Pa,w1∧TZ2Pw1,,w2∧…∧TZ2Pwk-1,b}infIZ2Pa,w1∨IZ2Pw1,,w2∨…∨IZ2Pwk-1,b}infFZ2Pa,w1∨FZ2Pw1,,w2∨…∨FZ2Pwk-1,b}

TZ2N(a,b),IZ2N(a,b),FZ2N(a,b)k=supTZ2Na,w1∨TZ2Nw1,,w2∨…∨TZ2Nwk-1,b}infIZ2Na,w1∧IZ2Nw1,,w2∧…∧IZ2Nwk-1,b}infFZ2Na,w1∧FZ2Nw1,,w2∧…∧FZ2Nwk-1,b}
(TZ2Pa, b,IZ2Pa, b,FZ2Pa, b)∞ and (TZ2Na, b,IZ2Na, b,FZ2Na, b)∞ is said to be the strength of connectedness between two vertices x and y in G, where

TZ2P(a,b),IZ2P(a,b),FZ2P(a,b)∞=supTZ2P(a,b), inf TZ2P(a,b), inf Fz2P(a,b)
TZ2N(a,b),IZ2N(a,b), FZ2N(a,b)∞=infTZ2N(a,b),supTZ2N(a,b),supFz2N(a,b)
If TZ2Pa, b≥TZ2Pa, b∞, IZ2Pa, b≤IZ2Pa, b∞, FZ2P(a, b)≤FZ2P(a, b)∞ and

TZ2Na, b≥TZ2Na, b∞, IZ2Na, b≤IZ2Na, b∞, FZ2N(a, b)≤FZ2N(a, b)∞ then the arc ab in G is called a strong arc. A path a-b is a strong path if all arcs on the path are strong.

BSVN detour periphery (Per<sub id="s-d99461422be5">BND</sub> (G)) and BSVN detour eccentric graph (Ecc<sub id="s-e7529900ee4c">BND</sub> (G))
Theorem 4.1 A BSVN graph G is a BSVN detour self-centered if and only if every node of G is a BSVN detour eccentric.

Proof. Suppose G is a BSVN detour self-centered BSVN graph and let b be a node in G.

Let a∈bBND*. So EccBNDb=BNDGa, b. Since G is a BSVN detour self-centered BSVN graph, EccBNDa=EccBNDb=BNDG(a, b) and this implies that b∈aBND*. Hence b is a BSVN detour eccentric node of G.

Conversely, let every node of G is a BSVN detour eccentric node. If possible, let G be not BSVN detour self centrad BSVN graph. Then radBNDG≠diamBND(G) and there exist node r∈G such that EccBNDr=diamBND(G). Also, let P∈rBND*. Let U be r-p BSVN detour in G. So there must have a node q on U for which the node q is not a BSVN detour eccentric node of U. Also, q cannot be a BSVN detour eccentric node of every other node. Again if q be a BSVN detour eccentric node of a node a (say), this means q∈aBND*. Then there exists an extension of a - q BSVN detour up to r or up to p. But this contradicts the facts that q∈aBND*. Hence, radBND (G) = diamBND (G) and G is a BSVN detour self-centered BSVN graph.

Theorem 4.2.
If G is a BSVN detour self-centered BSVN graph, then radBND G= diamBND G=n-1, where n is the number of nodes of G.

Proof. Let G be a BSVN detour self-centered BSVN graph.

If possible, let diamBND(G) = l < n - 1. Let U1 and U2 be two distinct BSVN detour peripheral path. Let p∈U1, q∈U2. So there exist a strong path between p and q, because of the connectedness of G. Then there exist nodes on U1 and U2, whose eccentricity > l, but this is impossible, because diamBND(G) = l. Hence U1 and U2 are not distinct. Since U1 and U2 are arbitrary, so there exist node r in G such that r is common in all BSVN detour peripheral paths. So EccBND(r)<l, which is impossible, because G is a BSVN detour self-centered. Hence, diamBND (G) = n - 1 = radBND (G).

Theorem 4.3.
For a connected BSVN graph G, PerBNDG=G if and only if the BSVN detour eccentricity of each node of G is n-1, n = number of nodes in G.

Proof. Let PerBNDG=G. Then EccBND(p) =diamBND (G), ∀ p∈G. So every node of G is a BSVN detour periphery node of G. Therefore G is a self-centered BSVN graph and radBND G= diamBND G= n-1. So the BSVN detour eccentricity of each node of G is n-1.

Conversely, let the BSVN detour eccentricity of each node of G is n-1. So, radBND G= diamBND G= n-1. All nodes of G are BSVN detour peripheral nodes and hence PerBND (G) = G.

Theorem 4.4.
For a connected BSVN graph G, EccBNDG=G if and only if the BSVN detour eccentricity of each node of G is n-1, n = number of nodes in G.

Proof. Let, EccBNDG=G, So all nodes of G are BSVN detour eccentric node. Therefore, G is a self-centered BSVN graph and diamBND (G) = n - 1. Hence, the BSVN detour eccentricity of each node of G is n - 1.

Conversely, let the BSVN detour eccentricity of each node of G is n-1. So radBNDG=diamBNDG=n-1. So all nodes of G are BSVN detour peripheral nodes as well as BSVN detour eccentric nodes. Hence, EccBND (G) =G.

Theorem 4.5.
In a connected BSVN graph G, a node b is a BSVN detour eccentric node if and only if b is a BSVN detour peripheral node.

Proof. Let b be a BSVN detour eccentric node of G and let b∈aBND*. Let x and y be two BSVN detour peripheral nodes, then BNDx, y=diamBNDG=k(say). Let P1 and P2 be any x - y and a - b BSVN detour in G respectively. There arise two cases.

Case 1: When b is not an internal node in G i.e, there is only one node, say c which is adjacent to b. So_{ } c∈P2. Since G is connected, c is connected to a node of P1, say c'. So either c'∈P2 or c'∈(P1∩P2). Thus, in any case the path from a to x or a to y through c and c' is longer than_{ } P2. But it is impossible since b is a BSVN detour eccentric node of a. Hence EccBNDa=diamBND(G) i.e, b is a BSVN detour peripheral node of G.

Case 2: When b is an internal node in G, then there exists a connection between b to x and b to y, because of the connectedness of G. Then a - b BSVN detour can be extended to x or y. This is impossible because b is a BSVN detour eccentric node of a. Hence EccBNDa=diamBNDG i.e, b is a BSVN detour peripheral node of G.

Conversely, we assume that b be a BSVN detour peripheral node G. So there exists a BSVN detour peripheral node say a (distinct from b). Therefore b is a BSVN detour eccentric node of a.