A New Approach to the Duplication of Blocks by a Vertex and AUM Block Mean Labeling with an Application in Water Management

Abstract

AUM block mean labeling is constructed from the labels of the vertices and edges. It is an interesting and rapidly developing labeling technique with a wide scope for applications in different fields. Duplication of vertices and edges is another direction of research among graph labeling techniques. In the present work, the notion of duplication of a block by a vertex and duplication of multiple blocks by a vertex are introduced and further discussed for Path Pn. Duplication of a block by a vertex means a new vertex is appended to the graph, and that vertex is joined to the vertices that are common to that block and its neighbouring blocks. Also, AUM block mean labeling is investigated for some graphs that are obtained by duplication of a block by a vertex. The concept of duplication of blocks by a vertex finds its application in water management. A new algorithm is presented by applying the concept of duplication of blocks, to facilitate a drainage system and water supply without blockages in this paper.