Sharp bounds on the distance spectral radius and the distance energy of graphs

Bounds for the distance Estrada index of graphs

“…Conversely, one can easily see that the equality holds in (14) for the complete graph K 2 . This completes the proof.…”

“…In [17], Zhou and Trinajstić showed that this situation is not valid for n ≥ 3 (see Theorem 4). Therefore the equality holds in (14) for only the graph G = K 2 .…”

“…Now we suppose that the equality holds in (14). Then all inequalities in the above argument must be equalities.…”

“…Moreover, the lower bounds in (7), (14) and (19) are nicer than the lower bound (4). For instance, let G be the path graph P 4 .…”

Remoteness and distance eigenvalues of a graph

“…Indulal [2] gave the following lower bound on the distance spectral radius of G. Let G(⌊ d+1 2 ⌋, ⌈ d+1 2 ⌉) denote by the graph obtained from K n−1−d by joining an endvertex of P ⌊ d+1 2 ⌋ and P ⌈ d+1 2 ⌉ to each vertex of K n−1−d . Zhang [6] determined that G(⌊ d+1 2 ⌋, ⌈ d+1 2 ⌉) attains the minimal distance spectral radius among all graphs with given diameter d and order n. Then we have the following result.…”