Now we will give some upper bounds for simple connected graphs. It is a result of Perron–Frobenius in matrix theory (see [7, p. 66]) which states that a non-negative matrix B always has a non-negative eigenvalu
1 we call a graph without isolated vertices -minimal if its th greatest Laplacian eigenvalue reaches this lower bound. We describe all 1-minimal and 2-minimal graphs and we prove that for every ? 3 the path k on + 1 vertices is the unique -minimal graph....
It is also known to be the unique tropical eigenvalue of A when it is irreducible [6]. The cycles i1i2…iki1 and nodes within said cycles, on which the maximum cycle mean λ(A) is attained, are called critical, and the subgraph of G(A) consisting of all nodes and arcs belonging...
Some new results are based on the entire degree sequence, e.g., [15]. The goal of this article is slightly shifted. We want to characterize connected graphs G that have greatest spectral radius in t...The spectra of some trees and bound for the largest eigenvalue of any tree, Linear ...