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Abstract

In a study of traffic, the labelling problems in graph theory can be used by considering the crowd at every junction as the weights of a vertex and expected average traffic in each street as the weight of the corresponding edge. If we assume the expected traffic at each street as the arithmetic mean of the weight of the end vertices, that causes mean labelling of the graph. When we consider a geometric mean instead of arithmetic mean in a large population of a city, the rate of growth of traffic in each street will be more accurate. The geometric mean labelling of graphs have been defined in which the edge labels may be assigned by either flooring function or ceiling function. In this, the readers will get some confusion in finding the edge labels which edge is assigned by flooring function and which edge is assigned by ceiling function. To avoid this confusion, we establish the F-geometric mean labelling on graphs by considering the edge labels obtained only from the flooring function. An F-Geometric mean labelling of a graph G with q edges, is an injective function from the vertex set of G to {1, 2, 3,..., q +1} such that the edge labels obtained from the flooring function of geometric mean of the vertex labels of the end vertices of each edge, are all distinct and the set of edge labels is {1, 2, 3,..., q}. A graph is said to be an F–Geometric mean graph if it admits an F–Geometric mean labelling. In this paper, we study the F-geometric meanness of the graphs such as cycle, star graph, complete graph, comb, ladder, triangular ladder, middle graph of path, the graphs obtained from duplicating arbitrary vertex by a vertex as well as arbitrary edge by an edge in the cycle and subdivision of comb and star graph.

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