The trace norm ‖G‖∗ of a graph G is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm ‖G‖∗ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note focuses on the maximum trace norm of r-partite graphs, which for r > 2 raises some unusual problems. It is shown that, if G is an r-partite graph of order n, ...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V1, V2, . . . , Vr} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k vertices in each vertex class by Ts,r(k). In this paper we prove that if h, r and s are positive i...

We study the minimum number of complete r-partite r-uniform hypergraphs needed to partition the edges of the complete r-uniform hypergraph on n vertices and we improve previous results of Alon.

For r 4 we determine the smallest number of vertices, .<71-( d), of complete that are decomposable into two isomorphic factors for a given finite diameter d. We also prove that for a ,d such graph exists for each order than gr( d). 1. INTRODUCTORY NOTES AND DEFINITIONS In this paper we study decompositions of finite complete multipartite graphs factors with prescribed diameter. A factor F of a ...

Using the modified tensor product of graphs defined in [1], we present here a graphical criterion for the separability of m-partite pure quantum states living in a real or complex Hilbert space. The criterion gives both necessary and sufficient condition for the separability of such states. This criterion was first conjectured by S. L. Braunstein, S. Ghosh, T. Mansour, S. Severini, and R. C. Wi...

The p-spectral radius of a uniform hypergraph G of order n is de…ned for every real number p 1 as (p) (G) = max jx1j + + jxnj = 1 r! X fi1;:::;irg2E(G) xi1 xir : It generalizes several hypergraph parameters, including the Lagrangian, the spectral radius, and the number of edges. The paper presents solutions to several extremal problems about the p-spectral radius of k-partite and k-chromatic hy...

Let K+ r (s1, . . . , sr) be the complete r-partite graph with parts of size s1 ≥ 2, s2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove that: (A) For all r ≥ 2 and all sufficiently small ε > 0, every graph of sufficiently large order n with tr (n) + 1 edges contains a K + r ( ⌊c lnn⌋ , . . . , ⌊c lnn⌋ , ⌈ ...