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Exact values of fractional dimensions for non-planar symmetric networks.

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Title: Exact values of fractional dimensions for non-planar symmetric networks.
Authors: Fatima, Arooba, Alkahtani, Badr S., Javaid, Muhammad, Shahab, Sana, Anjum, Mohd
Source: Maejo International Journal of Science & Technology; May-Aug2025, Vol. 19 Issue 2, p181-195, 15p
Subject Terms: GRAPH theory, ASYMPTOTIC analysis, COMPUTER networks, FRACTALS
Abstract: Fractional versions of graph-theoretic invariants have expanded their applicability to diverse fields such as connectivity, scheduling, assignment and operational research. Building on this extension of fractional graph theory, we introduce the local fractional metric dimension (LFMD) of non-planar networks. For a given network G, the local resolving neighbourhood LR(uw) of an edge uw is defined as the set of vertices in G that distinguish between u and w. A function ρ:V(G)→[0,1] is considered a local resolving function if ρ(LR(uw))≥1 for every edge uw in G. The LFMD of G is then defined as the minimum value of ρ(V(G)) taken over all possible local resolving functions. In this paper we compute the exact values of the LFMD for two non-planar, structurally symmetric network families--generalised gear and generalised helm networks--each constructed with a finite number of levels and characterised by distinct types of vertices. We compare the results for both networks and analyse the impact of pendant vertices in the context of bipartite and nonbipartite structures. Furthermore, we investigate the asymptotic behaviour as the order of the networks approaches infinity and present an emergency exit planning scenario to illustrate the practical significance of our findings. [ABSTRACT FROM AUTHOR]
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Abstract:Fractional versions of graph-theoretic invariants have expanded their applicability to diverse fields such as connectivity, scheduling, assignment and operational research. Building on this extension of fractional graph theory, we introduce the local fractional metric dimension (LFMD) of non-planar networks. For a given network G, the local resolving neighbourhood LR(uw) of an edge uw is defined as the set of vertices in G that distinguish between u and w. A function ρ:V(G)→[0,1] is considered a local resolving function if ρ(LR(uw))≥1 for every edge uw in G. The LFMD of G is then defined as the minimum value of ρ(V(G)) taken over all possible local resolving functions. In this paper we compute the exact values of the LFMD for two non-planar, structurally symmetric network families--generalised gear and generalised helm networks--each constructed with a finite number of levels and characterised by distinct types of vertices. We compare the results for both networks and analyse the impact of pendant vertices in the context of bipartite and nonbipartite structures. Furthermore, we investigate the asymptotic behaviour as the order of the networks approaches infinity and present an emergency exit planning scenario to illustrate the practical significance of our findings. [ABSTRACT FROM AUTHOR]
ISSN:19057873