Lit-only sigma game on a line graph

Let Γ be a connected graph. For any x ∈ F 2 V ( Γ ) , a move of the lit-only sigma game on Γ consists of choosing some v ∈ V ( Γ ) with x ( v ) = 1 and changing the values of x at all those neighbors of v . An orbit is a maximal subset of F 2 V ( Γ ) any two of which can reach each other by a series of moves. The minimum light number of Γ is max K min x ∈ K # supp ( x ) , where K runs through all orbits. by L ( Γ ) the line graph of Γ . The subspace of F 2 E ( Γ ) that is generated by the rows of the adjacency matrix of L ( Γ ) is dubbed σ 1 ( Γ ) . We view σ 1 ( Γ ) as a linear code in F 2 E ( Γ ) and let ρ ( σ 1 ( Γ ) ) be the covering radius of σ 1 ( Γ ) . For any S ⊆ V ( Γ ) , the edge-boundary of S is the set E B ( S ) of edges with exactly one endpoint lying inside S . The edge isoperimetric number of Γ , denoted b ( Γ ) , is defined to be max k min # S = k # E B ( S ) . in result of this note is that the minimum light number of L ( Γ ) is equal to max { ρ ( σ 1 ( Γ ) ) , b ( Γ ) } . We also determine the sizes of all orbits of the lit-only sigma game on L ( Γ ) . Especially, when Γ is a tree, we prove that the minimum light number of L ( Γ ) is b ( Γ ) . Moreover, if the tree Γ has n ≥ 3 vertices, the group formed by the moves of the lit-only sigma game on L ( Γ ) is shown to be the symmetric group on n elements.