On the energy and spread of the adjacency, Laplacian and signless Laplacian matrices of graphs (Q6586306)

Given the adjacency matrix \( A(G) \) of a graph \( G \), the diagonal matrix \( D(G) \) is obtained by taking the row sums of \( A(G) \). The Laplacian matrix of \( G \) is defined as \( L(G) = D(G) - A(G) \), while the signless Laplacian matrix is given by \( Q(G) = D(G) + A(G) \).\N\NThe authors establish relationships between the energy and spread of the adjacency, Laplacian, and signless Laplacian matrices of graphs. Specifically, they demonstrate that:\N\N-- the upper bound on the adjacency spread of a graph \( G \) provides a lower bound for the adjacency energy of \( G \);\N\N-- the upper bound on the adjacency energy of \( G \) provides a lower bound for the adjacency spread;\N\N-- the upper bound on the largest \( L \)-eigenvalue provides a lower bound for the Laplacian energy of \( G \);\N\N-- the upper bound on the Laplacian energy of \( G \) provides a lower bound for the largest \( L \)-eigenvalue;\N\N-- the upper bound on the signless Laplacian spread of \( G \) implies a lower bound for the Laplacian energy of \( G \);\N\N-- the upper bound on the Laplacian energy of \( G \) provides a lower bound for the signless Laplacian spread of \( G \).