Cyclic connectivity index of fuzzy incidence graphs with applications in the highway system of different cities to minimize road accidents and in a network of different computers

A parameter is a numerical factor whose values help us to identify a system. Connectivity parameters are essential in the analysis of connectivity of various kinds of networks. In graphs, the strength of a cycle is always one. But, in a fuzzy incidence graph (FIG), the strengths of cycles may vary even for a given pair of vertices. Cyclic reachability is an attribute that decides the overall connectedness of any network. In graph the cycle connectivity (CC) from vertex a to vertex b and from vertex b to vertex a is always one. In fuzzy graph (FG) the CC from vertex a to vertex b and from vertex b to vertex a is always same. But if someone is interested in finding CC from vertex a to an edge ab, then graphs and FGs cannot answer this question. Therefore, in this research article, we proposed the idea of CC for FIG. Because in FIG, we can find CC from vertex a to vertex b and also from vertex a to an edge ab. Also, we proposed the idea of CC of fuzzy incidence cycles (FICs) and complete fuzzy incidence graphs (CFIGs). The fuzzy incidence cyclic cut-vertex, fuzzy incidence cyclic bridge, and fuzzy incidence cyclic cut pair are established. A condition for CFIG to have fuzzy incidence cyclic cut-vertex is examined. Cyclic connectivity index and average cyclic connectivity index of FIG are also investigated. Three different types of vertices, such as cyclic connectivity increasing vertex, cyclically neutral vertex and, cyclic connectivity decreasing vertex, are also defined. The real-life applications of CC of FIG in a highway system of different cities to minimize road accidents and a computer network to find the best computers among all other computers are also provided.


Introduction
Graphs are convenient tools to explain associations between different types of entities under examination. Vertices or nodes denote entities, and edges or arcs explain the vertices' connections. A mathematical structure to describe unpredictability and equivocacy in daily life strong product for interval-valued FGs was provided by Rashmanlou, and Jun [27]. Sunitha and Vijayakumar [28] defined complement of a FG. Mordeson and Nair [29] introduced and examined the concepts of chords, twigs, 1-chains with boundary zero, cycle vectors, coboundary, and cocycles for FGs. They have also shown that although the set of cycle vectors, fuzzy cycle vectors, cocycles, and fuzzy cocycles do not necessarily form vector spaces over the field Z 2 of integers modulo 2, they nearly do. Later on, different mathematicians participated in the development of graphs and FGs. Their achievements can be seen in [30][31][32][33][34][35].
There is a flaw in FGs because they do not give any clue of the impact of a vertex on edge. This lack of FGs become the fundamental cause to establish the scheme of FIG. The proposal of FIGs was first initiated by Dinesh [36]. For example, in a highway system, if vertices represent various cites and edges serve as highways, introducing the degree of connection between city L and the highway LM joining cities L and M permits a profound analysis of the highway system. This connection could be the ramp system joining L and LM. We indicate this relationship by the ordered pair (L, LM). Malik et al. [37] applied FIGs in different types of applications. Mathew and Mordeson [38] proposed the idea of cut pairs and fuzzy incidence trees in FIGs. They also discussed some vital properties of FIGs. Three different types of nodes, including fuzzy incidence connectivity enhancing node, fuzzy incidence connectivity reducing node, and fuzzy incidence connectivity neutral node in FIGs was introduced by Fang et al. [39]. Like node and edge connectivity in graphs, Mathew et al. [40] discussed these concepts for FIGs. Mordeson and Mathew [41] developed fuzzy end nodes and fuzzy incidence cut vertices in FIGs. Nazeer et al. [42] presented the idea of intuitionistic fuzzy incidence graphs (IFIGs) as a generalization of FIGs along with their certain properties. They introduced a variety of operations in IFIGs. They also provided a fascinating application of the product of IFIGs. The idea of order, size, domination, strong fuzzy incidence domination, and weak fuzzy incidence domination in FIGs was proposed by Nazeer et al. [43]. Nazeer and Rashid [44] presented the idea of picture FIGs. They introduced picture fuzzy cut-vertices, picture fuzzy bridges, picture fuzzy incidence cut pairs, and picture fuzzy incidence cut-vertices. More extensive and comprehensive work on FIGs, can be seen [45][46][47][48].
Connectivity parameters are connectivity measures of any system. In graphs, the connectivity between any two vertices is 1, and in FGs, it is from closed interval [0, 1]. There are certain motives to propose the concept of CC in FIGs. Firstly, in FGs, we can only compute the CC from vertex l to vertex m and from vertex m to the vertex l but if someone is interested in examining the CC from vertex l to an edge lm, then FGs are not enough to answer this question. Therefore, we propose the concept of CC in FIGs because FIGs permit us to find the CC from vertex l to an edge lm due to the presence of an incidence pair in FIGs. Secondly, in FIGs, the CC from vertex l to an edge lm and vertex m to an edge lm may or may not be the same. Thirdly, we cannot apply graphs and FGs to the applications of the highway systems of different cities and networks of different computers provided in Section 5 due to the non-availability of the influence of a vertex on and edge. Fourthly, the objective to introduce these ideas to FIGs is that Mathew and Sunitha [16] initiated the notion of CC in FGs. Later, Binu et al. [49] initiated an idea of cyclic connectivity index (CCI) and average cyclic connectivity index (ACCI) of FGs. We extended their work for FIGs. This paper establishes CC, CCI and ACCI of FIGs.
The other part of this article is constructed as follows. Section 2 consists of some introductory outcomes essential to comprehend the remaining portion of the article. CC, fuzzy incidence cyclic cut-vertex (FICCV), fuzzy incidence cyclic bridge (FICB) and fuzzy incidence cyclic cut pair (FICCP) of FIG are explained in Section 3. The formula to determine CCI, the way to manipulate ACCI of FIG, and three different types of vertices, namely, cyclic connectivity increasing vertex (CCIV), cyclically neutral vertex (CNV), and cyclic connectivity decreasing vertex (CCDV) are described in Section 4. The real-life applications of CC of a FIG in a highway system of different cities to reduce road accidents and a computer network to find the best computers sharing the maximum amount of data among all other computers are discussed in Section 5. A comparative analysis of our study with the existing study is provided in Section 6. Section 7 carries some conclusions and future directions.

Preliminaries
This section carries some elementary and rudimentary definitions and results of FIGs. These will be useful to understand the contents of the article.^indicates the minimum operator, and _ denotes the maximum operator in this article. Definition 1. [41] A fuzzy subset (FSS) of a set is a function of the set into the closed interval Definition 3. [41] Let G = (V, E, I) be an IG. A sequence of distinct vertices P 1 : k 0 , (k 0 , k 0 k 1 ), k 0 k 1 , (k 1 , k 0 k 1 ), k 1 , . . ., k n−1 , (k n−1 , k n−1 k n ), k n−1 k n , (k n , k n−1 k n ), k n is called an incidence path and vertices k 0 and k n are said to be connected. The incidence strength (I s ) of P 1 is defined as η (k 0 , k 0 k 1 )^η(k 1 , k 0 k 1 )^. . .^η(k n−1 , k n−1 k n ) and is expressed by ðI s P 1 Þ. A sequence P 2 : k 0 , (k 0 , . ., k n−1 , (k n−1 , k n−1 k n ), k n−1 k n , (k n , k n−1 k n ), k n , (k n , k n k n+1 ), k n k n+1 is another incidence path between k 0 and k n k n+1 . The I s of P 2 is defined as

Cycle connectivity of fuzzy incidence graphs
In this section, we present the novel idea of connectivity named as Here inG, abca, abcda and adca are all FICs. There are three FICs passing through a and c comprising, abca, adca and abcda with I s =^{η(a, ab) = 0. 6 Next, we will propose a fascinating result related to FIC in the form of a proposition. We can easily calculate the O of any FIC by just applying this result. This proposition will help us to save time and energy. Also, this will be helpful to avoid very long calculations.
Proof. It follows by Proposition 1 that each I p is a I s p in a FIC. Therefore, the O of a FICG is the I s ofG. Now, we are going to introduce an actual result in the form of a theorem. With the help of this theorem, we will be able to compute O of any CFIG. By applying this theorem, we do not have to need to do complicated calculations. We have to use the theorem and get the required result.
Proof. Suppose the conditions of the Theorem. Since any three vertices ofG are adjacent becauseG is a CFIG also any three vertices are in 3 vertices FIC. SinceG is a CFIG, by Proposition 2 all I p are I s p in CFIG. Therefore, to calculate the smallest I s of FIC inG, it is enough to calculate the smallest I s of every 3 vertices FIC inG. SinceG is a CFIG therefore to examine a 4 vertices FIC, C = abcda inG (it is notable that the case is same for n vertices FIC) there will be parts of two 3 vertices FIC in C, namely C 1 = abca and Consider, η(a, ac) = j, then I s (C 1 ) = I s (C 2 ) = I s (C) = j. Suppose η(a, ac) > j, since I s (C) = j then either C 1 or C 2 will have I s equal to j. Now, I s (C) =^{I s (C 1 ), I s (C 2 )}. Thus the I s of a 4 vertices FIC is same as the I s of a 3 vertices FIC inG. From all 3 vertices FIC, the 3 vertices FIC devised by three vertices with largest vertices strength will have the greatest strength. Therefore, the FIC

Definition 15. A vertex l in a FIGG s said to be a FICCV if
OðG À lÞ < OðGÞ:

Definition 16. An edge (l, m) in a FIGG is said to be a FICB if
OðG À ðl; mÞÞ < OðGÞ:

PLOS ONE
To show that k n−3 < k n−2 . Assume that k n−3 = k n−2 . Then C 1 = h n h n−1 h n−2 and C 2 = h n h n−1 h n−3 have the equal I s , and hence the deleting of h n−2 , h n−1 or h n−3 will not lessen OðGÞ. This contradiction shows that k n−3 < k n−2 .
Conversely, assume that k n−3 < k n−2 . Now, we have to show thatG has a FICCV. Since k n−2 � k n−1 � . . . � k n and k n−3 < k n−2 , all FICs ofG have I s less than that of I s of h n−2 h n−1 h n−3 . Hence the removal of h n−1 , h n−2 or h n−3 will become the cause of reduction of OðGÞ therefore, G has a FICCV.    Proof. By given statement of theorem.G is a CFIG with |σ � | = n � 3 therefore Proposition 2 yields thatG will be without any δ − IPr. This means all I p inG are I s p and as stated in Proposition 1 every FIC is a strong FIC inG. Suppose l and m are any two vertices ofG then we have to calculate I s of all FICs contain vertices l and m. After this, we have to compute O which is the maximum value of I s of all FICs containing pair of vertices l and m. Similarly, we have to compute O up to n (total number of vertices) and take the minimum value of all O of the CFIGG. Also, the total number of edges for a CFIG is always equal to n

PLOS ONE
Hence from Eqs (1) and (2) it can be concluded that Here, we are going to present a very foundational concept of ACCI of a FIG. In enormous networks, the sturdy flow among different vertices is mandatory to sustain trustability and devotedness. To guarantee the firmness of the exchange of data in the complete or portion of the network, measuring the average value of the cyclic data exchange is vital. Therefore, we discuss the ACCI of a FIG. The ACCI of any FIG is denoted by  In a FIG an isolated vertex is 1 : a, b, d, a, C 2 : b, c, d, c and C 3 : a, b, c, d, a with I

Real-life applications of cycle connectivity
In daily life, O has various uses. Here, we have proposed two critical real-life applications of O of FIGs. In the first application, we have taken a highway system of different cities and apply the idea of O of FIG to find the roads which are becoming the leading cause of maximum accidents. In the second application, we have taken a network of different computers sharing data. We have applied the idea of O to the network of different computers and find which computer/computers are transferring the maximum amount of data to other computers.

Application of cycle connectivity in highway system
Due to the huge traffic on roads, the percentage of accidents is increasing day by day. To minimize these accidents government should take some serious steps to lessen the percentage of road accidents. Here, we are presenting a graphical model of Thus OðGÞ ¼ 0:3 is representing that the roads joining cities c 1 c 2 , c 1 c 8 and c 2 c 8 are the main roads which are becoming a main cause of highest percentage of road accidents. So, the government should focus on these roads by making more speed breakers, speed bumps and deploying more traffic wardens on these roads. In this way, they can minimize the percentage of road accidents.
We have used FIGs in our application. The FIGs are more instrumental and effective than graphs. We cannot use graphs to explain the above phenomenon because graphs do not show the impact of a vertex on an edge. Another thing, in graphs, the O between each pair of vertices is always equal to 1, and we are unable to find which roads are becoming the main cause of maximum road accidents, but in FIGs, the O between each pair of vertices will be different. Therefore, FIGs are more helpful and useful than graphs.

Application of cycle connectivity in a computer network
In a network of different computers, computers are sharing data with each other. We want to find which computer/computers are best in performance among all other computers and sharing maximum data with all other computers in a network. This can be done by computing O between each pair of computers in a network. The pair of computers which have a maximum O will be the required computers transferring maximum data to all other computers in a network. Here, we are presenting a graphical model of FIG to explain this phenomenon. As an example, assume a network of FIG comprising of eight vertices. The vertices are showing the eight distinct computers in a network. The MSV of the vertices is indicating data store in each of these computers, the MSV of the edges is demonstrating the total amount of data that can be transferred from one computer to another computer and the MSV of the I p is representing the amount of data which one computer is transferring to another computer. For example, an I p (a, ab) is indicating the transfer of data from computer a to computer b and an I p (b, ab) is showing the transfer of data from computer b to computer a. LetG ;h ¼ 0:01 and OG g;h ¼ 0:2. Thus OG g;h ¼ 0:2 is representing the maximum O between computers g and h. Therefore, computers g and h are best computers in performance among all other computers and sharing maximum data with all other computers in a network.

Comparative analysis
Here, we are going to compare our model with the existing model. In Fig 9 a FIG is indicating  a Since in case of graph the O between each pair of vertices is equal to 1. Therefore, we are unable to find the roads which are becoming a main reason of maximum accidents. Hence our model is better than the previous one.
Similarly, in Fig 10 a FIG is representing a network of different computers. Computers are sharing data. We want to find which computer/computers are best in performance among all other computers and sharing maximum data with all other computers in a network. This can be done by computing O between each pair of computers in a network. The pair of computers which have a maximum O will be the required computers transferring maximum data to all other computers in a network. The vertices are showing the eight distinct computers in a network. The MSV of the vertices is indicating data store in each of these computers, the MSV of the edges is demonstrating the total amount of data that can be transferred from one computer to another computer and the MSV of the I p is representing the amount of data which one computer is transferring to another computer. In the case of graph the O between each pair of vertices is OG a; Therefore, pervious model is not helpful to find which computer/computers are transferring maximum amount of data. Thus, our model is more effective and beneficial than the previous one.

Conclusion
In this article, we advanced the theory of FIGs. The notion of connectivity is indivisible from the theory of FIGs. There are a variety of parameters that command the connectivity of a network. In this article, the authors attempted to make up a new connectivity idea named as O, the best computers among all other computers in a network is also provided. A comparative analysis of our study with the existing study is also provided. More related ideas will be contemplated in the upcoming papers.