Scholarly work in the Department of Mathematics & Statistics

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Browsing Scholarly work in the Department of Mathematics & Statistics by Title

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    Bounds of the Radio Number of Stacked-Book Graph with Odd Paths
    (arXiv preprint, 2022) Adefokun Tayo Charles; Ajayi Deborah Olayide
    A Stacked-book graph Gm,n is obtained from the Cartesian product of a star graph Sm and a path Pn, where m and s are the orders of the star graph and the path respectively. Obtaining the radio number of a graph is a rigorous process, which is dependent on diameter of G and positive difference of non-negative integer labels f(u) and f(v) assigned to any two u, v in the vertex set V (G) of G. This paper obtains tight upper and lower bounds of the radio number of Gm,n where the path Pn has an odd order. The case where Pn has an even order has been investigated.
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    Distance Two Labeling of Direct Product of Paths and Cycles
    (arXiv preprint, 2013) Adefokun Tayo Charles; Ajayi Deborah Olayide
    Suppose that [n] = {0, 1, 2, ..., n} is a set of non-negative integers and h, k ∈ [n]. The L(h, k)-labeling of graph G is the function l : V (G) → [n] such that |l(u) − l(v)| ≥ h if the distance d(u, v) between u and v is one and |l(u) − l(v)| ≥ k if the distance d(u, v) is two. Let L(V (G)) = {l(v) : v ∈ V (G)} and let p be the maximum value of L(V (G)). Then p is called k h−number of G if p is the least possible member of [n] such that G maintains an L(h, k)−labeling. In this paper, we establish 1 1− numbers of Pm × Cn graphs for all m ≥ 2 and n ≥ 3.
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    Generalised Radio Number of Odd Paths
    (Crawford Journal of Natural & Applied Sciences, 2019) Adefokun Tayo Charles
    Generalised Radio labelling of a graph G, an aspect signal assignment problem, and which generalises radio labelling problem, is a non-negative integer function on V (G) of G to the set of positive integer, such that for u, vV (G), | g(u)  g(v) |  diam(G)  d(u, v)  k, with diam(G) being the diameter of G and d(u,v), the distance between vertices u and v. The optimum value of span(g) , which the difference between the largest and the smallest radio label on V(G) is the radio number rng(G) of G. This work obtains the generalised radio number for the odd path Pn.
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    Hamiltonian Complete Number of Some Variants of Caterpillar Graphs
    (arXiv preprint, 2022) Adefokun Tayo Charles; Ajayi Deborah Olayide; Ogundipe Opeoluwa Lawrence; Onaiwu Kingsley Nosa
    A graph G is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are adjacent. For a non-Hamiltonian graph G, the Hamiltonian complete number H(G) is the minimum number of edges that must be added to G to make it Hamiltonian. We focus on both regular and irregular caterpillar graphs, deriving explicit formulas for H(G) in various cases. Specifically, we show that for a regular caterpillar graph Gn(k) where each vertex on the central path is adjacent to k leaves, H(Gn(k)) = n(k−1). We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for H(G) in these cases. Our results contribute to the understanding of Hamiltonian properties in tree-like structures and have potential applications in network design and optimization.
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    On Bounds of Radio Number of Certain Product Graphs
    (Journal of the Nigerian Mathematical Society, 2018) Adefokun Tayo Charles; Ajayi Deborah Olayide
    Given a graph G, whose vertex set is V (G), the radio labelling of G is a variation of vertex labelling of G which satisfy the condition that given any v1, v2 ∈ V (G), and some positive integer function f(v) on V (G), then |f(v1) − f(v2)| ≥ diam(G) + 1 − d(v1, v2). Radio labelling guarantees a better reduction in interference in signal-dependent networks since no two vertex have the same label. The radio number rn(G) of G is the smallest possible value of f(v) such that for any other vk ∈ V (G), f(vk) < f(v). In this work, we consider a Cartesian product graph obtained from a star and a path and determined upper and lower bounds of the radio number for the family of these graphs.
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    On Induced Matching Numbers of Stacked-Book Graphs
    (International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2024) Adefokun Tayo Charles; Ogundipe Opeoluwa Lawrence; Ajayi Deborah Olayide
    For a simple undirected graph G, an induced matching in G is a set of edges M no two of which have common vertex or are joined by an edge of G in the edge set E(G) of G. Denoted by im(G), the maximum cardinal number of M is known as the induced matching number of G. In this work, we probe im(G) where G = Gm,n, which is the stacked-book graph obtained by the Cartesian product of the star graph Sm and path Pn.
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    On Maximum Induced Matching Numbers of Special Grids
    (arXiv preprint arXiv, 2016) Adefokun Tayo Charles; Ajayi Deborah Olayide
    A subset M of the edge set of a graph G is an induced matching of G if given any two e1, e2 ∈ M, none of the vertices on e1 is adjacent to any of the vertices on e2. Suppose that MIMG, a positive integer, is the largest possible size of M in G, then, M is the maximum induced matching, MIM, of G and MIMG is the maximum induced matching number of G. We obtain some upper bounds for the maximum induced matching numbers of some specific grids
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    On Radio Number of Stacked-Book Graphs
    (arXiv preprint arXiv, 2019) Adefokun Tayo Charles; Ajayi Deborah Olayide
    A Stacked-book graph Gm,n results from the Cartesian product of a star graph Sm and path Pn, where m and n are the orders of Sm and Pn respectively. A radio labeling problem of a simple and connected graph, G, involves a non- negative integer function f : V (G) → Z+ on the vertex set V (G) of G, such that for all u, v ∈ V (G), |f(u) − f(v)| ≥ diam(G) + 1 − d(u, v), where diam(G) is the diameter of G and d(u, v) is the shortest distance between u and v. Suppose that fmin and fmax are the respective least and largest values of f on V (G), then, spanf, the absolute difference of fmin and fmax, is the span of f while the radio number rn(G) of G is the least value of spanf over all the possible radio labels on V (G). In this paper, we obtain the radio number for the stacked-book graph Gm,n where m ≥ 4 and n is even, and obtain bounds for m = 3 which improves existing upper and lower bounds for Gm,n where m = 3.