Department of Mathematics

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Author: search.filters.author.Dr. Hafiz Muhammad Afzal SiddiquiSubject: search.filters.subject.SP20

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    Secure Resolving Dimension of Zero Divisor Graphs Associated to Commutative Rings
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Muhammad Muzammil; CIIT/SP20-RMT-012/LHR; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 7626
    This Thesis contains the study of secure resolving dimensions of certain zero divisor graphs of commutative ring with unity. A subset 𝑇 of 𝐺 is a secure resolving set for 𝐺 if 𝑇 is resolving and for any 𝑥 ∈ 𝑉 − 𝑇, there exists 𝑦 ∈ 𝑇 such that (𝑇 − {𝑦}) ∪ {𝑥} is a resolving set for 𝐺. The minimum cardinality of a secure resolving set for 𝐺 is known as the secure resolving dimension for 𝐺. We determine the secure resolving dimension of various zero divisor graphs derived from commutative ring with unity and prove that all these graphs have constant secure resolving dimension
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    Zero Forcing Number of F-Sum of Some Families of Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Aleeha Shahid; CIIT/SP20-RMT-020/LHR; Dr. Hafiz Muhammad Afzal Siddiqui; LHR TP 7651
    This thesis deals with zero forcing number of 𝐹-sum of graphs derived from certain families of graph. A zero forcing set of graph 𝑆 is a set 𝑆 ⊆ 𝑉 such that there are no white vertices in the derived coloring of 𝑆. The zero forcing number of 𝐺 is 𝑍(𝐺) = min⁡{|𝑆|: 𝑆⁡is zero forcing set of⁡𝐺}. We study zero forcing number of 𝐹-sum of two path graph, two cycle graph and then path and cycle graph. It has been proved that it is constant in case of 𝑆 operation of 𝐹-sum and unbounded otherwise.