Study of Convexo-Symmetric Networks via Fractional Dimensions

For having an in-depth study and analysis of various network’s structural properties such as interconnection, extensibility, availability, centralization, vulnerability and reliability, we require distance based graph theoretic parameters. Numerous parameters like distance based dimensions help in designing queuing models in restaurants, public health and service facilities and production lines. Likewise, allocations of robots in several production units are indebted to these. Moreover, chemists and druggists use these parameters in finding out new drug type and structural formula of a chemical compound. In this article, we are finding out the extremal values of local fractional metric dimension of a class of network bearing convex as well as symmetric properties known by the name of convex polytopes. Also we have related the significance of our findings towards the end of the manuscript in the form of fire exit plan. The same will serve as a guide for future architects in planning a floor with a conducive fire exit plan.

. Type I Convex Polytope B n for 1 ≤ j ≤ n s j the vertices of inner cycle, t j the vertices of internal cycle, u j the vertices of external cycle and v j the vertices of outer cycle. effective usage [12], [23]. Similarly, the allocation of robots 100 in some production unit as well as in public health facility 101 have been given attention in [27]. Also, numerous techniques 102 that involve the rectification of example and picture han-103 dling, information handling, see [26]. The aforementioned 104 parameters have made it possible for engineers and scientists 105 to employ robots to serve a production line, public health 106 facilities and restaurants with the least number of these but 107 also minimized their responding time. is known as local minimal resolving function if any function 118 ψ : V (G) → [0, 1] such that ψ ≤ κ and ψ(c) = κ(c) for at 119 least one c ∈ V (G), that is not a resolving function of G. 120 For a connected network G its LFMD is given by 121 dim lf (G) = min{|κ| : κ is the minimal LRF of G} [31].

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The network G ∼ = B n bearing n 4-sided faces, 3n 3-sided 124 faces, n 5-sided faces and a pair of n-sided faces that is the 125 resultant of the combination of convex polytope Q n and prism 126 network D n [17] is known as type I convex polytope. Its vertex 127 and edge sets are: The induced cycles namely inner, internal, external and outer 132 cycle are {v j |1 ≤ j ≤ n} and {v j |1 ≤ j ≤ n} respectively. The Figure 1 134 illustrates B n .

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Likewise, by adding new edges v j+1 v j in B n we obtain the 136 Type II Convex Polytope network G ∼ = C n . It consists of n 137 4-sided faces, n 5-sided faces, 3n 3-sided faces, and a pair of 138 VOLUME 10, 2022 FIGURE 2. Type II Convex Polytope C n for 1 ≤ j ≤ n s j the vertices of inner cycle, t j the vertices of internal cycle, u j the vertices of external cycle and v j the vertices of outer cycle.
n-sided faces. In the same manner, the cycles {s j |1 ≤ j ≤ n}, j ≤ n} are known as inner, internal, external and outer cycle 141 respectively. The Figure 2 illustrates C n .

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The sets V (C n ) and E(C n ) are given by 146 We close this section by presenting 2 important results that 147 will be required to prove our findings. We are presenting here   another resolving function g such that g ≤ f . By definition, which shows that our supposition was wrong about g being 174 the LRF. Hence, f is the minimal LRF. Suppose thatf as 175 another LRF of G. Consider, In this way, the following 3 subcases comes into play: In the light of aforementioned subcases, equation 1 takes 182 the form: Finally, from both Case I and Case II, we get    Then, 1 < dim lf (B n ) ≤ 10n 7n+2 .
respectively. We note that     The LRNs are given by:

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It can be seen that Table 7, Table 8 and Table 9 show  Table 6 bears LRNs with the minimum cardinality of 21 and 350    Likewise, in Table 10 shows the LRNs with maximum cardi-359 nality of 30 = γ hence by Theorem B and Corollary C (G is 360 not bipartite), we have Therefore, from 6 and 7, we have Case II: n ≥ 8.

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• Table 11 shows the summary of main results and 388 Table 12 gives the values of LFMDs as they tends to ∞. 389 • The graphical analysis of results as shown in Figure 3 390 reveals that B n bears the dominant LFMD.

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• Based on the findings of our results, the fire exit plan 392 for the floor of building can be designed. For instance, 393 consider a floor of the mall comprising of shops, food 394 court, washrooms, lifts and escalators as shown on the 395 top left of Figure 4. As a first step its graph theoretic 396 version is obtained as shown on the top right of Fig-397 ure 4. Whereas at the center of Figure 4 we obtain the 398 underlying graph as C 6 . From Theorem 4.4 the LFMD 399 of C n is 30/21 ≈ 1.43 which means that the complete 400 evacuation for this floor in case of fire will be done in 401 1 hr and 43 min. Similarly, the path for the fire exit plan 402 can be found from