Monophonic Cover Pebbling Number \((MCPN)\) of Network Graphs

Given a connected graph \(G\) and a configuration \(D\) of pebbles on the vertices of \(G\), a pebbling transformation involves removing two pebbles from one vertex and placing one pebble on its adjacent vertex. A monophonic path is defined as a chordless path between two non-adjacent vertices \(u\) and \(v\). The monophonic cover pebbling number, \(\gamma_{\mu}(G)\), is the minimum number of pebbles required to ensure that, after a series of pebbling transformations using monophonic paths, all vertices of \(G\) are covered with at least one pebble each. In this paper, we determine the monophonic cover pebbling number (\(MCPN\)) for the gear graph, sunflower planar graph, sun graph, closed sun graph, tadpole graph, lollipop graph, double star-path graph, and a class of fuses.