Sharp lower bounds for Coulomb energy

Quasimodes and Resonances: Sharp Lower Bounds

We prove that, asymptotically, any cluster of quasimodes close to each other approximates at least the same number of resonances, counting multiplicities. As a consequence, we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes.

Some Lower Bounds for Estrada Index

Sharp lower bounds for the Zagreb indices of unicyclic graphs

The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which th...

Sharp Bounds on the PI Spectral Radius

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.

Sharp lower bounds on the fractional matching number

A fractional matching of a graph G is a function f from E(G) to the interval [0, 1] such that ∑ e∈Γ(v) f(e) ≤ 1 for each v ∈ V (G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α′∗(G), is the maximum of