European Journal of Combinatorics

European Journal of Combinatorics

Two conjectures on the cospectrality of complete tripartite graphs are false

For a graph G of order n with adjacency eigenvalues lambda_1(G)>= ...>= lambda_n(G), the cospectrality of G within the set G of all graphs of order n is the smallest spectral distance from G to another graph of G. Abdollahi and Zakeri conjectured that for every n>= 4 the cospectrality of the complete tripartite graph K...

Fractional clique decompositions in random hypergraphs

We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic er...

On the Stability of the Independence Number of the Distance Graph G(n,r,

We consider a random subgraph G_p(n,r,<s) of the complete distance graph G(n,r,<s) whose vertices are the r-element subsets of the set {1,...,n} and whose edges join pairs of subsets that intersect in fewer than s elements; each edge survives independently of the others with probability p. The stability of the Erdos-Ko...

Journal of Number Theory

The largest α-spectral radius of the k-uniform unicyclic hypergraphs with perfect matchings

Let G be a k-uniform hypergraph with k ⩾ 2 and 0 ⩽ α < 1. The α-spectral radius of G is the largest modulus of all the eigenvalues of Aα (G), where Aα (G) = αD(G) + (1-α)A(G) is the convex linear combination of D(G) and A(G) with D(G) and A(G) being the degree diagonal tensor and the adjacency tensor of G, respectively...