Symmetric Shannon capacity is the independence number minus 1

A symmetric variant of the Shannon capacity of graphs is defined and computed.

Generalizations and Strengthenings of Ryser's Conjecture

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

On graphs with minimal distance signless Laplacian energy

For a simple connected graph G of order n having distance signless Laplacian eigenvalues ρ 1 Q ≥ ρ 2 Q ≥ ⋯ ≥ ρ n Q \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as D S L E ( G ) = ∑ i = 1 n | ρ i Q - 2 W ( G ) n | DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|} where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t), 1 ≤ t ≤ ⌊ n - k 2 ⌋ 1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.