University of Zimbabwe Institutional RepositoryThe UZ eScholar digital repository system captures, stores, indexes, preserves, and distributes digital research material.http://irr.uz.ac.zw:80802021-03-02T15:17:40Z2021-03-02T15:17:40ZAn Upper Bound on the Radius of a 3-Vertex-Connected C4-Free GraphFundikwa, Blessings T.Mazorodze, Jaya P.Mukwembi, Simonhttps://hdl.handle.net/10646/39642021-02-15T09:16:21Z2020-08-04T00:00:00ZAn Upper Bound on the Radius of a 3-Vertex-Connected C4-Free Graph
Fundikwa, Blessings T.; Mazorodze, Jaya P.; Mukwembi, Simon
Let G=(V,E) be a finite, connected, undirected graph with vertex set V and edge set E. -e distance dG (u,v) between two vertices u, v of G is the length of a shortest u-v path in G.-e eccentricity ec(v)of a vertex v∈V is the maximum distance between v and any other vertex in G. -e value of the minimum eccentricity of the vertices of G is called the radius of G denoted by rad(G). -e degree deg(v)of a vertex v of G is the number of edges incident with v. -e minimum degree δ(G)is the minimum of the degrees of vertices in G.-e open neighbourhood N(v)of a vertex v is the set of all vertices of G adjacent to v. -e closed neighbourhood N[v]of v is the set N(v)∪v{ }. A graph is triangle-free if it does not contain C3 as a subgraph and C4−free if it does not contain C4 as a subgraph. For notions not defined, here we refer the reader to [1].
The results in this paper are part of the first author’s MPhilSc thesis.
2020-08-04T00:00:00ZUpper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge ConnectivityFundikwa, Blessings T.Mazorodze, Jaya P.Mukwembi, Simonhttps://hdl.handle.net/10646/39632021-02-15T09:11:58Z2020-09-03T00:00:00ZUpper Bounds on the Diameter of Bipartite and Triangle-Free Graphs with Prescribed Edge Connectivity
Fundikwa, Blessings T.; Mazorodze, Jaya P.; Mukwembi, Simon
Graph theory is used to study the mathematical structures of pairwise relations among objects. Mathematically, a pair G=(V,E) is a crisp graph, where V is a nonempty set and E is a relation on V[1]. -e order of a graph G is the number of vertices of G and is denoted by|V|=n. -e size of G,denoted by|E|=m, is the number of edges of G. -e distance,d G(u,v), between two vertices u,v of G is the length of a shortest u-v path in G. -e eccentricity, ec G(v), of a vertex v∈V is the maximum distance between v and any other vertex in G. -e maximum distance among all pairs of vertices [2], also known as the value of the maximum ec-centricity of the vertices of G, is called the diameter of G denoted by diam(G). -e degree, deg(v), of a vertex v of G is the number of edges incident with v. -e minimum degree,δ(G)=δ, of G is the minimum of the degrees of vertices in G. -e open neighborhood,N(v), of a vertex v is simply the set containing all the vertices adjacent to v. -e closed neighborhood, N[v], of a vertex v is simply the set containing the vertex v itself and all the vertices adjacent to v.We denote by E(V1,V2)the set {e=xy|x∈V1,y∈V2}of edges with one end inV1and the other end inV2.-e edge connectivity, λ(G)=λ, of G is the minimum number of edges whose deletion from G results in a disconnected or trivial graph. A complete graph,Kn, is a graph in which every vertex is adjacent to every other vertex. -e most likely antonym for a complete graph is a null graph,Nn, which is a graph containing only vertices and no edges. A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge in G connects a vertex in U to a vertex in V; furthermore, no two vertices in the same set are adjacent to each other. A graph is triangle-free if it does not contain C3 as a subgraphandC4-free if it does not containC4as a subgraph. It is important to observe from the above definitions that every bipartite graph is triangle free, but there are some triangle-free graphs which are not bipartite, for example, a cycle graph with five vertices(C5). For notions not defined here,we refer the reader to [3].
2020-09-03T00:00:00ZSpanning paths in graphsMafuta, PhillipMukwembi, SimonMunyira, Sheunesuhttps://hdl.handle.net/10646/39622021-02-15T09:08:42Z2018-08-29T00:00:00ZSpanning paths in graphs
Mafuta, Phillip; Mukwembi, Simon; Munyira, Sheunesu
The Conjecture, Graffiti.pc 190, of the computer program Graffiti.pc, instructed by DeLavi˜na,
state that every simple connected graph G with minimum degree δ and leaf number L(G)
such that δ ≥ 1 2 (L(G) + 1), is traceable. Here, we prove a sufficient condition for a graph
to be traceable based on minimum degree and leaf number, by settling completely, the
Conjecture Graffiti. pc 190. We construct infinite graphs to show that our results are best
in a sense. All graphs considered are simple. That is, they neither have loops nor multiple
edges.
2018-08-29T00:00:00ZVirtual Memorial Service for the Late Professor Rosemary Moyana, PVC Academic AffairsUniversity of Zimbabwe, Media Teamhttps://hdl.handle.net/10646/39612021-02-12T13:21:09Z2021-02-12T00:00:00ZVirtual Memorial Service for the Late Professor Rosemary Moyana, PVC Academic Affairs
University of Zimbabwe, Media Team
2021-02-12T00:00:00Z