Department of Mathematics
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Item Computing Metric Dimension Of Certain Families(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2019) By: Wajiha Nazir,; SP17-RMT-021; Contributor(s): Dr. Muhammad Faisal Nadeem, Assistant Profesor [Supervisor]; LHR TP 4636Computing Metric Dimension of Certain Families of Toeplitz Graphs Let 𝑄 = {𝑞1,𝑞2,…𝑞𝑘} be an ordered set of vertices of a graph 𝐺 and 𝑎 is any vertex in 𝐺, then 𝑎 has representation w.r.t 𝑄, denoted by 𝑟(𝑎|𝑄) is the 𝑘 −tuple which is (𝑟(𝑎,𝑞1),𝑟(𝑎,𝑞2)…𝑟(𝑎,𝑞𝑘)). If the different vertices of 𝐺 have the different representation w.r.t 𝑄, then 𝑄 is known as a resolving set/locating set. A resolving/locating set having the least count of vertices is basis of 𝐺 and count of vertices in this basis is called metric dimension of 𝐺 which is represented as 𝑑𝑖𝑚(𝐺). In our work, we study the metric dimension of certain Toeplitz graphs and find out their metric dimension. Firstly, we find the metric dimension of Toeplitz graph � �𝑛〈1,𝑡〉 where 𝑡 ≥ 2, which is constant. Secondly, we find the metric dimension of Toeplitz graph 𝑇𝑛〈1, 𝑠, 𝑡〉, where 𝑡 = 3,4,5,6 and 𝑠 = 2, which is also constant. ixItem On Characterization Of Spanning Forest Complexes(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2016) Muhammad Asif,; FA14-MSMAT-004; Dr. Hani Shaker, Assistant Profesor [Supervisor]; LHR TP 4636In this thesis, we discuss the algebraic and combinatorial aspects of spanning forest complex ∆s(G) associated to simple disconnected graph G. We mainly emphasis on disjoint union of isomorphic copies graph G and provide special importance to tCn where tCn is the disjoint union of t cycles of length n and call this graph t-cycle graph. We give the characterization of s(tCn), all spanning forests, of tCn. We intend to provide the algebraic and combinatorial characterization of spanning forest complex ∆s(tCn) associated to tCn. In particular, we compute the formula for the f-vector of spanning forest complex of t-cycle graph. We also compute the h-vector and Hilbert series of Stanley Riesner ring k[∆s(tCn)]