Metric – based fractional dimensions of connected graphs
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Date
2023-01-26
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UMT Lahore
Abstract
Let g = (v (g), e(g)) be a graph with v (g) and e(g) ⊆ v (g) × v (g) as the sets of vertices and edges respectively. A walk is defined as a sequence of alternating vertices and edges. A path between any two vertices of g is a walk without repetition of any vertex. For any two vertices a, b ∈ v (g), the distance d(a, b) is the length of shortest path between them in g. A graph is said to be connected if there exists a path between any pair of vertices of g. A vertex x ∈ v (g) is said to resolve {a, b} ⊆ v (g) if d(x, a) ̸ = d(x, b). Let s = {v1, v2, v3, ..., vn} ⊆ v (g) be an ordered set and x ∈ v (g), then the n tuple representation of x with respect to s is d(x|s) = ((x, v1), (x, v2), (x, v3), ..., (x, vn)). If the distinct vertices of g have distinct representations with respect to s then s is called resolving (locating) set. The resolving set with minimum cardinality is called the metric basis of g and the cardinality of metric basis is known as metric dimension of g (dim(g)). For a pair a, b ∈ v (g), the resolving neighbourhood set of g is defined as r(a, b) = {z ∈ v (g) : d(z, a) ̸ = d(z, b)}. A real valued function f : v(g)→ [0, 1] such that f (r(a, b)) ≥ 1 for each r(a, b) is called a resolving function of g, where f (r(ab)) = p x∈lr(ab) f (x). A resolving function f is called minimal if there exists a function g : v (g) → [0, 1] such that g ≤ f and g(x) ̸ = f (x) for at least one x ∈ g that is not resolving function of g. The fractional dimension of g is donated by dimf (g) and defined as dimf (g) = min{|f | : f is minimal resolving function of g}. Furthermore, if, we replace the pair of vertices (a, b) ∈ v (g) with and edge ab ∈ e(g) then resolving function becomes local resolving function and fractional metric dimension becomes local fractional metric dimension. Slater (1975) defined the idea of the resolving (locating) sets for a connected graph to find the reference or location number for a connected graph. Malter and harary studied the same concept but they call it different name known as metric dimension. In (2006) fahar et al. introduced the idea of fractional metric dimension to improve the solution of cretin integer programming problem. Arguman et al. (2012) devaloped computational techniques to compute fractional metric dimension of different connected graphs. Aisyah et al. (2019) introduced the latest derived form of fractional metric dimension called by local fractional metric dimension of connected graphs. In this thesis for all the connected graphs, a general combinatorial technique is established to improve the lower bound of local fractional metric dimension from unity. In the same context, the local fractional metric dimension of prism-related graphs such as circular diagonal ladder, antiprism, and sun flower graphs are computed with the help of obtained criteria. In addition exact 8 value and sharp bounds for different types of generalized convex polytope graphs, generalized gear graphs, sunlet graphs( middle sunlet , total sunlet), toeplitz graphs are also computed. For all the obtained results, the bounded and unboundedness of local fractional metric dimension are also presented when the order of these graphs approaches to infinity. Furthermore, fractional metric dimension of prism - related graphs (circular diaginal ladder, double sun flower) and double path graph is obtained in the form of exact values. For all the obtained results, the bounded and un- bundedness of lfmd and fmd are also presented when order of all the graphs approaches to infinity.