Research Article Connectivity and Wiener Index of Fuzzy Incidence Graphs

Connectivity is a key theory in fuzzy incidence graphs ( FIGs ) . In this paper, we introduced connectivity index ( CI ) , average connectivity index ( ACI ) , and Wiener index ( WI ) of FIGs. Three types of nodes including fuzzy incidence connectivity enhancing node ( FICEN ) , fuzzy incidence connectivity reducing node ( FICRN ) , and fuzzy incidence connectivity neutral node ( FICNN ) are also discussed in this paper. A correspondence between WI and CI of a FIG is also computed.


Introduction and Preliminaries
Zadeh [1] presented the theory of fuzzy set (FS) to resolve complications in tackling with precariousness. Since then, the FS theory becomes a rich area in multiple disciplines, including mathematics, computer science, and signal processing. e theory of graphs has been considered to play a vital role in dealing with real-life situations. A graph is an easy way of expressing information, including the relationship between different objects. e objects are shown by nodes, and relations are represented by edges. In this paper, all graphs are finite, simple, without loops, and undirected. When there is a lack of certainty in the illustration of the objects and their association, we need to draw a fuzzy graph (FG) model. Zadeh's FS provided a productive ground for the theory of FGs which has been proposed by Rosenfeld [2]. In a graph, the strength of connectedness (SC) between any two vertices is either 0 or 1, whereas in FG, it is a real number ∈∈[0, 1]. e study of FGs leads many scientists to contribute in this field, such as Yeh and Bang [3] studied the concept of FGs independently and discussed its applications in clustering analysis. Bhattacharya and Suraweera [4] discussed an algorithm to compute the max-min powers and property of FGs. Bhutani [5] worked on automorphism of FGs. Mordeson [6] introduced fuzzy line graphs. Bhutani and Rosenfeld [7,8] studied strong arcs as well as fuzzy end nodes in FGs. Sunitha and Vijayakumar [9,10] defined fuzzy trees and fuzzy blocks in FGs. Samanta et al. [11] inaugurated completeness and regularity of generalized FGs. Samanta and Pal [12] studied fuzzy planner graphs. Mathew and Sunitha [13] classified the edges of a FG as an α − strong, β − strong, and δ − edge. Mathew and Sunitha [14,15] presented vertex, edge connectivity, and cycle connectivity in FGs. Mathew et al. [16] initiated saturation in FGs, and Binu et al. [17] explored CI and its application in FGs. Binu et al. [18] investigated CI of FG and its application to human trafficking. For some other significant works on graphs and FGs, one may refer to [19][20][21][22][23][24][25].
Wiener [26] was the first who investigated WI when he was studying about the boiling point of paraffin. After the landmark work of Harold Wiener about WI, in the middle of 1970s, new results related to WI were described. In graphs, WI has been studied in different fields such as Chemistry, Mathematics, and Physics. Binu et al. [27] discussed WI of FG and its application to illegal immigration networks.
FGs are unable to provide any information on the effect of a vertex on edges of the graph. erefore, this disadvantage opens a way to introduce FIGs. FIGs talk about the effect of a vertex on an edge. Dinesh [28] presented the idea of FIGs. For example, if vertices show different residence societies and edges show roads joining these residence societies, we can have a FG expressing the extent of traffic from one society to another. e society has the maximum number of residents and will have maximum ramps in society. So, if c and d are two societies and c d is a road joining them, then (c, c d) could express the ramp system from the road c d to the society c. In the case of an unweighted graph, c and d both will have an influence of 1 on c d. In a directed graph, the influence of c on c d represented by (c, c d) is 1, whereas (d, c d) is 0. is idea can be generalized by FIGs. Mordeson [29] studied numerous connectivity perceptions in FIGs. Malik et al. [30] explained different uses of FIGs. Mordeson et al. [31] proposed a fuzzy incidence (HTML translation failed) blocks along with their applications. Mordeson and Mathew [32] discussed different connectivity ideas in FIGs. e motivation of our work is that CI, ACI, and WI of FGs exists in literature, but these indices are unknown for FIGs. ese indices will make a way to study different properties of FIGs at length. is is why we propose these concepts for FIGs. Our work will open the new doors for many researchers to study FIGs in detail. e outline of this paper is as follows. In Section 1, we provide elementary definitions, results, and expressions of FIGs, which are required for the development of the content. In Section 2, we discuss CI of FIGs. Section 3 describes certain boundaries for CI of FIGs. CI of vertex and edge deleted fuzzy incidence subgraph (FIS) is illustrated in Section 4. Section 5 explains ACI and its characteristics. In Section 6, we discuss WI of FIG and a relationship between connectivity and WI. Below, we present some preliminary definitions from [17,19,32].
Let G be a simple graph with vertex set V(G) and edge set E(G). en, an incidence graph (IG) is given by G � (V, E, I), where I ⊆ V × E. An IG is shown in Figure 1, and if (u, uv) is in IG, then (u, uv) is said to be an incidence pair or pair. Assume an IG G � (V, E, I).
, v n is said to be a walk. It is closed if v 0 � v n . A walk is called a path if it has all distinct vertices. An IG is said to be connected if all pair of vertices are joined by a path. An edge ab is said to be a fuzzy bridge (FB) if the deletion of ab ∈ θ * lessens the SC between some pair of vertices in G.
In this paper, minimum is represented by ∧ and maximum is expressed by ∨. Definition 1. Consider a graph G � (V, E), and η and θ are fuzzy subsets of V and E, respectively. Assume V × E has a fuzzy subset ψ. If ψ(v * , e * ) ≤ η(v * )∧θ(e * ) for every v * ∈ V and e * ∈ E, then ψ is called a FI of G.
A FI path λ from g to h, g, gh ∈ η * ∪ θ * , is defined as a sequence of elements η * , θ * , and ψ * beginning with g and closing with h. e In FIG, the incidence paths (IPs) can take distinct forms. ab) shows the highest IS of a − ab. If ψ(a, ab) > ICONN G− (a,ab) (a, ab), then the pair is called α − strong. If ψ(a, ab) � ICONN G− (a,ab) (a, ab), then this type of pair is β − strong. If an incidence pair is α − strong or β − strong, then this kind of pair is a strong pair. If HTML translation failed, then this type of pair is called δ − incidence pair.
Definition 9 (see [19]). e distance d(u, v) between two vertices u, v ∈ V(G) is the minimum number of edges in a path between u and v in G. Definition 10. In a graph G, a path of shortest length is called geodesic.
Definition 11. (see [19]). WI of a graph G is the sum of distances between all pairs of vertices of G. en, the WI of a graph G is given by

Connectivity Index of Fuzzy Incidence Graph
Connectivity is a common parameter associated with a network. is section includes the introduction and formula to calculate CI of FIG. For easiness, in the coming sections, we will take η(a) � 1 for every a ∈ η * .
is the maximum value of ISs for all the possible IPs between a and b.
e connectivity indices of subgraphs of FIGs can never be surpassed that of the FIGs. erefore, a subgraph H of FIG G will have to be less than or equal to CI than the CI(G).
is is shown in the coming proposition. Figure 3 having

Bounds for Connectivity Index of Fuzzy
Incidence Graph is section discusses some bounds for the CI of FIGs. Every  FIG has a different CI. erefore, all FIGs have different bounds for the CI. From all FIGs, the complete FIGs will have the highest CI. It is shown in the next theorem.  ″ (a, b).
en, ψ(a, ab) > ICONN G− (a,ab) (a, b). is means an edge ab is the only strongest path whose strength is equal to ψ(a, ab). From this, it is obvious that CI(G * ) < CI(G). As by definition α, strong arcs are FBs.
is means if CI(G * ) < CI(G), then ab is a FB. is shows that CI(G * ) < CI(G) iff uv is a FB.
Definition 14. Assume a FIG and z ∈  In the following proposition with the help of CI, we classify these nodes: z be FICNN of FIG. en, ACI(G

Proof. Let
which implies CI(G)/CI(G − z) � n/(n− 2). By reversing the argument, the sufficient part can easily be proved. In similar manners, the other two cases can be solved.

Wiener Index of FIG
where d s (a, b) represents weights of those strong geodesics from a − b whose sum is minimum.
Example 7. Consider G be a FIG given in Figure 7 with Link between WI and CI of a FIG In FIGs, it could be noted that CI will be less than WI. Figure 9 with

Example 9. Consider a FIG given in
us, CI(G) < WI(G).  a and b is d s (a, b), whereas the minimum membership value of all strong incidence pairs is Mathematical Problems in Engineering ICONN G (a, b), s (a, b). Hence, CI(G) < WI(G).  a with b is d s (a, b), whereas the minimum membership value of incidence pair s (a, b).

Conclusion
Connectivity is an essential parameter attached to a network. e idea of connectivity is inseparable from the theory of FIGs. In this paper, we have come up with different results about WI and CI of FIGs. Relevant examples related to WI and CI of FIGs are too obtained. In this article, CI, ACI, and WI of FIGs linked with networks are expressed. Nodes of FIGs are classified as FICRN, FICEN, and FICNN by using these incidences. Various types of FIGs are also obtained. A crucial relationship between CI and WI of FIG is derived too. Our objective is to enlarge our research work to soft FIGs, bipolar FIGs, threshold FIGs, competition FIGs, regular FIGs, and q-rung FIGs. More similar results and applications will be reported in upcoming papers.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.  Mathematical Problems in Engineering