We prove that for all values of the edge probability p(n) the largest eigenvalue of a random graph G(n, p) satisfies almost surely: λ1(G) = (1 + o(1))max{ √ ∆, np}, where ∆ is a maximal degree of G, and the o(1) term tends to zero as max{ √ ∆, np} tends to infinity.

In this paper, we present some inequalities for the Co-PI index involving the some topological indices, the number of vertices and edges, and the maximum degree. After that, we give a result for trees. In addition, we give some inequalities for the largest eigenvalue of the Co-PI matrix of G.

In this paper, we show that if the second largest eigenvalue of a d-regular graph is less than d − 2(k−1) d+1 , then the graph is k-edge-connected. When k is 2 or 3, we prove stronger results. Let ρ(d) denote the largest root of x3 − (d− 3)x2 − (3d− 2)x− 2 = 0. We show that if the second largest eigenvalue of a d-regular graph G is less than ρ(d), then G is 2-edge-connected and we prove that if...

We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.

We characterize the simple connected graphs with second largest eigenvalue less than 1/2, which consists of 13 classes specific graphs. These hint that c2∈[1/2,2+5], where c2 is minimum real number c for every greater a limit point in set eigenvalues leave it as problem.