Abstract
A pebbling move on a graphGconsists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graphG, denoted byf(G), is the leastnsuch that any distribution ofnpebbles onGallows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. This paper determines the pebbling numbers and the 2-pebbling property of the middle graph of fan graphs.
Highlights
Pebbling on graphs was first introduced by Chung [1]
We introduce some definitions and give some lemmas, which will be used in subsequent proofs
In [9], Ye et al proved that f(M(C2n)) = 2n+1 + 2n − 2 (n ≥ 2) and f(M(C2n+1)) = ⌊2n+3/3⌋ + 2n, where M(Cn) denotes the middle graph of Cn
Summary
Pebbling on graphs was first introduced by Chung [1]. Consider a connected graph with a fixed number of pebbles distributed on its vertices. For a distribution of pebbles on G, denote by p(H) and p(V) the number of pebbles on a subgraph H of G and the number of pebbles on a vertex V of G, respectively. In [9], Ye et al proved that f(M(C2n)) = 2n+1 + 2n − 2 (n ≥ 2) and f(M(C2n+1)) = ⌊2n+3/3⌋ + 2n, where M(Cn) denotes the middle graph of Cn. Motivated by these works, we will determine the value of the pebbling number and the 2-property of middle graphs of Fn. 2. The subgraph induced by S is a complete graph with n vertices.
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