On the construction of l-equienergetic graphs

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For a graph with n vertices and m edges, having Laplacian spectrum μ₁, μ₂,,μₙ and signless Laplacian spectrum μ+₁,μ+₂,,μ+ₙ, the Laplacian energy and signless Laplacian energy of G are respectively, defined as LE(G)=∑ᵢ₌₁ⁿ|μᵢ-2m/n| and LE+(G)=∑ᵢ₌₁ⁿ|μ+ᵢ-2m/n|. Two graphs G₁ and G₂ of same order are said to be L-equienergetic if LE(G₁)=LE(G₂) and Q-equienergetic if LE⁺(G₁)=LE⁺(G₂). The problem of constructing graphs having same Laplacian energy has been considered by Stevanovic for threshold graphs and by Liu and Liu for those graphs whose order is n≡ 0 (mod 7). In general the problem of constructing L-equienergetic graphs from any pair of given graphs is still not solved, and this work is an attempt in that direction. We construct sequences of non-cospectral (Laplacian, signless Laplacian) L-equienergetic and Q-equienergetic graphs from any pair of graphs having same number of vertices and edges.