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A graph related to the Euler ø function

Published online by Cambridge University Press:  03 July 2023

Nima Ghanbari  and
Saeid Alikhani
Nima Ghanbari
Affiliation:
Department of Informatics University of Bergen P.O. Box 7803, 5020 Bergen, Norway e-mail: [email protected]
Saeid Alikhani
Affiliation:
Department of Mathematics, Yazd University. 89195-741, Yazd, Iran e-mail: [email protected]
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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G, denoted by d (u, ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1].

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Information
The Mathematical Gazette , Volume 107 , Issue 569 , July 2023 , pp. 263 - 272
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Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association

References

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