Benue Journal Of Mathematics & Mathematics Education A Publication Of Benue State Branch Of The Mathematical Association Of Nigeria Vol.2 No.21 October 2021 ISSN 2141-8780 Editor-In-Chief Prof. Benjamin Imobo Imoko MMAN Department Of Science and Mathematics Education, Faculty Of Education, Benue State University, Makurdi © Benue State Branch Of The Mathematical Association Of Nigeria Citation: This Journal shall be cited as: Benue Journal Of Mathematics & Mathematics Education: A Publication Of Benue State Branch Of The Mathematical Association Of Nigeria, Vol.2 No.21 October 2021 ISSN 2141-8780 published & Printed by: Gee Tigons Enterprises, 12, ejoor Str., near Modern Market, Makurdi. electronic version on: http://groups.google.com/group/benue-journal-of-mathematics-mathematics-education Benue State Branch Of The Mathematical Association Of Nigeria Executive Council Members Chairman: E. A. Sev MMAN Vice Chairman: Dr D. A. Anjila Secretary General: Dr C. O. Igyu Assistant Secretary General: Z. F. Niba Treasurer: J. Ugbagir Financial Secretary: O. A. Ochihu Publicity Secretary: Z. N. Iyornum EX-Officio I: P.P. Iamegh EX-Officio II: K. Gire Immediate Past Chairman: R. V. Tsebo MMAN Editorial Board Members Editor-In-Chief: Prof. B. I. Imoko MMAN Benue State University, Makurdi Deputy Editor-In-Chief: Dr. Adikwu, O. Federal University of Agriculture, Makurdi Editors: Prof. Iji, C. O. Federal University of Agriculture, Makurdi Dr. Kwen, A. A. College of Education, Katsina Ala Mr. Abuul, A. A. College of Education, Oju Mr. Tsebo, R. V. MMAN College of Education, Katsina Ala Managing Editor Dr. Atovigba, M. V. MMAN Benue State University, Makurdi Editorial Consultants Prof. Kimbir, A. R. Federal University of Agriculture, Makurdi Prof. Onah, S. E. National Mathematical Centre, Abuja Call for papers This journal publishes well researched and peer reviewed articles (theoretical and empirical) on pure and applied mathematics, mathematics education, statistics and science. The journal is published quarterly. Authors may send their papers as follows. *Each paper must be typed double line spacing on A4 paper size of not more than 12 pages. The abstract should not be more than 150 words, typed on a separate sheet of paper. Authors are to write their names in full with surname coming first. GSM and email addresses should be included on the front page for easy communication. Citation and referencing should follow the most recent APA or any standard format. A neatly typed electronic copy of the manuscript should be sent to mikeatovigba@gmail.com. *Peer review of each article is concluded within two weeks of receipt, and authors whose articles have been accepted for publication should send: (1) A soft copy of the corrected article (on Microsoft word) to: mikeatovigba@gmail.com. (2) The assessment and publication fee of N10,000.00 to the following account: Name of Account: Michael Atovigba Account Number: 1000207732 Keystone Bank PLC, Makurdi. *All articles accepted for publication in Benue Journal of Mathematics and Mathematics Education (BJMME) become the copyright of Mathematical Association of Nigeria Benue State Branch. However, authors shall be held liable for any copy right violation and plagiarism in their works. *A hard copy of the published journal will be forwarded to the author for each article published. Authors may however request for additional copies at the cost of N500.00 each copy. For more clarification and correspondence, and for subscription, contact: Prof. Prof. B. I. Imoko PhD, MMAN Editor-In-Chief +2348060574236. Table of Contents 1 2 Metric Dimension of the Tensor Product of an Odd Length Cycle Graph and a Cycle Graph of Order 5 - O. H. Tyav & G. G. Apav A Path to Certainty - M. V. Atovigba & M. M. Chianson-Akaa 5-8 9-10 Metric Dimension of the Tensor Product of an Odd Length Cycle Graph and a Cycle Graph of Order 5 Orya Herbert Tyav & Godwin Gema Apav Department of Mathematics, College of Education, Katsina-Ala, Benue State Nigeria Corresponding Author: oryaherberttyav@gmail.com Abstract Let 𝑐𝑛 be an odd length cycle graph of order 5, having adjacency bases and the cardinality of the adjacency basis equals the adjacency dimension. The work examines the metric dimension of the tensor product of odd cycle graph and a graph of order 5, in which two adjacency bases W1 and W2 of C5 are used to obtain the resolving set of the tensor product. The metric dimension of the tensor product of cycle graphs have been obtained in terms of the order of 𝐶𝑚 and adjacency dimension of C5 and we have deduced the formula for finding metric dimension of a tensor product of odd length cycle graph and a graph of orders, C5. Key Words and Phrases Cycle Graph of Order 5, Odd Length Cycle Graph, Metric Dimension, Tensor Product 1.0 Introduction Distance related concepts are commonly used in graphs. Some of these concepts are diameter, girth, metric and dimension. The metric dimension of a graph G is a minimum cardinality resolving set for that graph. A graph can have more than one resolving set but the concern is on the minimum cardinality resolving set. The concept of resolving set is not only an early idea in graph but also a subject that has been studied by many graph theorists. It was first studied by Harary and Melter (1976). Product graphs are graph operations which combine two graphs to obtain one graph. The tensor product of circle graph Cm and Cn consists of two sets, the vertex set and edge set. The edge set connects the vertices together. The vertices, in this product graph are Cartesian products of the set. Any two vertices (𝑢1, 𝑢2) and (𝑣1, 𝑣2) in a tensor product of graph are adjacent if and only if 𝑢1𝑣1 ∈ Cm and 𝑢2𝑣2 ∈ Cn. The tensor product of graphs is one of the known product graphs. The tensor product of an odd length cycle graph and an odd length cycle graph consists of one component while the tensor product of an odd length cycle graph and an even length cycle graph consists of two components. Also, the tensor product of an odd length cycle graph and a graph of Order 5 has a single component. Finding the parameters of products of graphs is one of the common problems in graphs. The metric dimension of the tensor product of odd cycle graph and a graph of order 5 has two adjacency bases W1 and W2 of C5 used in obtaining the resolving set of the tensor product. The metric dimension of the tensor product of cycle graphs, on the other hand, is obtained in terms of the order of 𝐶𝑚 and adjacency dimension of C5. Amraei (2015) has studied the metric dimension of the tensor product of a complete and complete graph while Saputro et al (2013) have studied the metric dimension of a connected graph and a disconnected graph. In another related work, Moshen and Benhaz (2012) have studied the lexicographic product of graphs in terms of order of the graph and adjacency dimension. In this work, we are concerned with finding the metric dimension of the tensor product of an odd length cycle graph and a cycle graph of order 5. 2.0 Method In this work, we give some elementary results which will be used in establishing results on adjacency basis and metric dimension of tensor product of graphs. We shall start with the following definitions. 2.1 Preliminaries Definition 2.1.1: Adjacency resolving set: let be a graph, and a 𝑊 = {𝑤1, … . , 𝑤𝑘}𝑉(𝐺). For each vertex 𝑣 ∈ (𝐺), the adjacency representation of with respect to is the - vector Where (𝑣, 𝑤𝑖) = { 0 𝑖𝑓 𝑣 = 𝑤𝑖 1 𝑖𝑓 𝑣 ~ 𝑤𝑖 2 𝑖𝑓 𝑣 ≁ 𝑤𝑖 The set is an adjacency resolving set for G if the vectors are distinct. Definition 2.1.2: Minimum cardinality of an adjacency resolving set: The minimum cardinality of an adjacency resolving set is the adjacency basis of , denoted by 𝜇̂(𝐺). An adjacency resolving set of cardinality𝜇̂(𝐺) is an adjacency dimension of . Definition 2.1.3: Complement of a graph: If is a simple graph with vertex set (𝐺), it complement𝐺̅ is the simple graph with vertex set (𝐺)in which two vertices are adjacent if and only if they are not adjacent in . Definition 2.1.4: Path/Cycle: A Path is a walk in which all edges and all the vertices are different. A cycle is a closed walk in which all the edges are different and all the intermediate vertices are also different. Next we give a proposition on adjacency basis and metric basis of a tensor product of an odd length cycle graph. Proposition 2.2.1: Given the graph 𝐶𝑚 ⊕ 𝐶5, where m ≥ 3 is a positive integer.If there exists two adjacency bases 𝑊1 and 𝑊2 of 𝐶5 such that there is no vertex with adjacency representation 1 with respect to 𝑊1 and no vertex with adjacency representation 2 with respect to 𝑊2 then µ(Cm ⊕ C5) = m𝜇̂(C5) Proof: Let vi ∈ V(Cm) and vj ∈ V(C5) By vertex Vij ∈ (Cm ⊕ C5), we mean the jth vertex in the ith row of (Cm ⊕ C5). Let 𝑆 be a subset of (Cm ⊕ C5). We need to show that: µ(Cm ⊕ C5) = m𝜇̂(C5). Suppose 𝑊1 and 𝑊2 are adjacency bases of C5 such that (j/W1) ≠ 1 and (j/W2) ≠ 2. Obtain the metric basis of Cm ⊕ C5 as follows; S = (vij ∈ (Cm ⊕ C5): i ≡ 1 (mod2) ∈ W1) ∪ (vij ∈ (Cm ⊕ C5): i ≡ 0 (mod2), j ∈ W2) The vertices in 𝑆 resolve the graph Cm ⊕ C5 and it is minimum. We have µ(Cm ⊕ C5) = S. since covers the whole range of and belongs to adjacency bases of C5 Therefore 𝑆 = m𝜇̂(C5). Hence µ(Cm ⊕ C5) = m𝜇̂(C5) 3.0 Results A graph is used to illustrate the proposition stated in this work. This establishes some of the mathematical facts included in the proposition. The following example illustrates proposition 2.2.1 Example 1 Compute the metric dimension of the tensor product of 𝐶7 ⊕ 𝐶5 Solution From proposition 2.2.1, we have 𝑖 = (1,3,5,7), 𝑊1 = (1,4) and 𝑖 = (2,4,6), 𝑊2 = (2,3) S = (vij ∈ (Cm ⊕ C5): i ≡ 1 (mod2), ∈ W1) ∪ (vij ∈ (Cm ⊕ C5): i ≡ 0 (mod2),j ∈ W2). Therefore, S = ((1,1),(1,4),(2,2),(2,3),(3,1),(3,4),(4,2),(4,3),(5,1),(5,4),(6,2),(6,3),(7,1),(7,4)) The metric representation of vertices in the graph are given below r(1,1/S) = (0,2,1,3,2,2,3,3,4,3,4,2,5,3) r(1,2/S) = (4,2,5,1,4,2,4,3,3,3,2,4,1,3) r(1,3/S) = (2,4,2,4,2,4,3,4,4,3,4,2,3,1) r(1,4/S) = (2,0,3,1,4,2,4,3,3,4,2,4,3,5) r(1,5/S) = (4,4,3,3,4,4,4,3,3,4,2,2,1,1) r(2,1/S) = (6,3,4,2,6,3,4,2,4,3,3,3,2,2) r(2,2/S) = (1,3,0,4,1,3,2,4,3,4,4,3,4,2) r(2,3/S) = (3,1,4,0,3,1,4,2,3,3,3,4,2,4) r(2,4/S) = (3,6,2,4,3,6,2,4,3,4,2,3,2,2) r(2,5/S) = (1,1,2,2,1,1,2,2,4,3,3,3,4,4) r(3,1/S) = (2,2,1,3,0,2,1,3,2,2,3,4,4,4) r(3,2/S) = (4,2,6,1,4,2,6,1,4,2,4,3,3,4) r(3,3/S) = (2,4,6,1,2,4,1,6,2,4,3,4,3,3) r(3,4/S) = (4,2,3,1,2,0,3,1,4,2,4,5,3,1) r(3,5/S) = (4,4,3,3,4,4,3,3,4,4,3,3,3,3) r(4,1/S) = (4,3,4,2,6,3,4,2,5,3,4,2,4,3) r(4,2/S) = (3,4,2,4,1,3,0,4,1,3,2,4,3,3) r(4,3/S) = (4,5,4,2,3,1,4,0,3,1,4,2,4,3) r(4,4/S) = (3,4,2,4,3,5,2,4,3,5,2,4,3,4) r(4,5/S) = (3,3,2,2,1,1,2,2,1,1,2,2,4,3) r(5,1/S) = (4,3,3,4,2,2,1,3,0,2,1,3,2,2) r(5,2/S) = (3,4,4,3,4,2,6,1,4,2,6,1,4,2) r(5,3/S) = (4,3,3,4,2,4,1,5,2,4,1,5,2,4) r(5,4/S) = (4,4,3,3,4,2,3,1,2,0,3,1,4,2) r(5,5/S) = (4,3,3,4,2,4,1,5,2,4,1,5,2,4) r(6,1/S) = (2,2,3,3,4,3,4,2,5,3,4,2,5,3) r(6,2/S) = (4,2,4,3,3,4,2,4,1,3,0,4,1,3) r(6,3/S) = (4,4,3,4,3,3,4,2,3,1,4,0,3,1) r(6,4/S) = (2,2,3,3,3,4,2,4,3,5,2,4,3,6) r(6,5/S) = (4,4,3,3,4,3,2,2,1,1,2,2,1,1) r(7,1/S) = (5,3,4,2,4,3,3,3,2,2,1,3,0,2) r(7,2/S) = (1,3,2,4,3,4,4,3,4,2,5,1,4,2) r(7,3/S) = (3,1,4,2,3,3,3,4,2,4,1,5,2,4) r(7,4/S) = (3,5,2,4,3,4,4,3,2,2,3,1,2,0) r(7,5/S) = (1,1,2,2,4,3,3,4,4,4,3,3,4,5) Figure 6: The graph of C7C5 4.0 Discussion The metric dimension of the tensor product of an odd length cycle graph and a cycle graph of Order 5 denoted by Cm ⊕ C5 is obtained in terms of the order of Cm and adjacency basis of C5. This implies that to obtain a resolving set S, for a tensor product of odd length cycle graph and a cycle graph of Order 5, we first obtain the adjacency basis of C5 and then, the metric basis of the tensor product graphs. For the tensor product graphs Cm ⊕ C5 where C5 has adjacency basis W1 and W2 such that there is no vertex with adjacency representation 1 with respect to W1 and there is no vertex with adjacency representation 2 with respect to W2, We obtain the metric basis of 𝐶7 ⊕ 𝐶5 as S = ((1,1),(1,4),(2,2),(2,3),(3,1),(3,4),(4,2),(4,3),(5,1),(5,4),(6,2),(6,3),(7,1),(7,4)). Its minimum cardinality is 14, which is its metric dimension. Of note is that µ(𝐶7 ⊕ 𝐶5) = 7 𝑥 𝜇̂(𝐶5) = 7 𝑥 2 = 14 5.0 Conclusion In this work, we have developed a formula for finding the metric dimension of a tensor product of an odd length cycle graph C𝑚 and a cycle graph of order 5, C5 in terms of order of the graph C𝑚 and adjacency basis of C5. This is done by combining the vertices of the odd length cycle with vertices of the adjacency basis of C5 to generate a resolving set for the graph. References Amraei, H., Maimani, H. R., Selfy, A. O. & Zaeembashi, A. (2015). The metric dimension of the tensor product of ciques. Accessible at https://arXiv:1505.05811v1 Harary, F. & Melter, R. A. (1976). On the metric dimension of a graph. Ars Combintoria, 2:191-195 Moshen, J. and Benhaz, O. (2012). The metric dimension of the lexicographic product of graphs. Journal of Discrete Mathematics, 3(12):3348-3356 Saputro, S. W., Suprijanto, D., Baskoro, D. & Salman, A. M. N. (2013). The metric dimension of the lexicographic product of graphs. Journal of Discrete Mathematics,313: 10451051. A Path to Certainty Michael Vershima Atovigba PhD & Martha Mimi Chianson-Akaa PhD Department of Science and Mathematics Education Benue State University Makurdi Corresponding Author: mikeatovigba@gmail.com, matovigba@bsum.ng Abstract The work looks at probability of dependent events occurring as a route for traveling to the state of certainty. Thus the work demonstrates this path to certainty which is the probability value of 1. A formula is proposed and proved for walking towards certainty. Key Words and Phrases Certainty, Conditional Probability, Dependent Events Introduction The probability of an event is a number between 0 and 1, where 0 indicates impossibility of the event and 1 indicates certainty (Hygot Technologies, 2020). Two events are dependent if the outcome of the first event affects the outcome of the second event (OnlineMathLearning, 2020). When then, is certainty expected to occur in a probabilistic experiment? How would a search be conducted in order to arrive at a definite expected or desired result of finding the object desired? These are the posers that have motivated this study. For very often a community or a system desires to obtain some specific object in the community or system in view and the issue is calculating the probability that one object selected or tried among available objects is the one desired. It happens that the first object out of number may not be the desired, and the second or the chosen too. In that case it ends up being the last object in the array that would be the case in the selection thus lending the search for certainty a dependent or conditional probability. Method Conditional probability is the method of the study. By conditional or dependent probability is meant a previous event affects the chance of the next event that would happen in a given probability experiment (Laerd Statistics, 2017). The study hinges on this definition, that is: for a container of objects, the first object to be successfully selected has the probability of and once the object in the container is selected it is not returned into the container, so what number of objects left in the container is and the probability of the next object to be selected is for an array where is a natural number. Results The results of the study are in form of a proposition. A proof of the proposition is attempted. Proposition The more successive events in a dependent probability experiment the closer to certainty is the next probability value. Proof Let an array number objects in a container be listed such that there is an array where is a natural number. The first object to be selected has the probability of being successfully picked to be . By conditional probability the next object to be selected has the probability of being successfully picked to be . Thus the object would be picked at the probability where is the number of objects already picked. Using , it becomes obvious that which is certainty. For: . Discussion and Conclusion Thus in a conditional probability experiment, the more successive selections that are made, if the previous ones were not the desired, the higher the chances are that the next object selected would be the desired. Thus any search machine like Yahoo or Google can use this formula in finding what is being searched for. The behaviour of such a search machine would be the algorithm: Declared: is an integer, is real: Input ; Output ; Stop: if such that is the desired object. In any system where alternatives are available it is often required to select the most suitable alternative that might produce the most useful results. Such could use this formula in selecting. For instance, in a learning environment there might be methods of learning a topic and the challenge is to select the most suitable method. In that case an array of the methods is made such that each one selected and tried has the probability of being the suitable method given by . Whenever a particular method fails to produce the desired learning experience the next available method is selected until possibly the method, and that is just what has been espoused, that is, which is certainty. Such an approach is recommended to the teacher in presenting lessons such that if one method or approach does not yield satisfactory results or learning experiences then that lesson should be re-planned and re-presented using the next available method or approach until the learning desired is realized or achieved. The same route is followed if possible methods of production in an industry are at play. Thus more progressive selections successively made in a conditional probability experiment portend higher chances (closer to certainty) that the next object selected would be the desired object. In any system where alternatives are available, to select the most suitable alternative that might produce the most useful results requires use of such an algorithm. References Hygot Technologies (2020). Probability. Accessible at www.toppr.com Laerd Statistics (2017). Understanding descriptive statistics and inferential statistics. Accessible at: https://statistics.laerd.com/statistical-guides/descriptive-inferential-statistics.php OnlineMathLearning (2020). Probability of dependent events. Accessible at www.onlinemathlearning.com Benue Journal of Mathematics and Mathematics Education 2(21) October 2021 10