We construct an efficient randomized algorithm that, given a digraph, tests if it is a strongly connected graph. And the query complexity of our algorithm is polylogarithmic in the size of the digraph.

Let H be a digraph possibly with loops and let D be a digraph whose arcs are colored with the vertices of H (an H-colored digraph). A walk (path) P in D will be called an H-restricted walk (path) if the colors displayed on the arcs of P form a walk in H. An H-restricted kernel N is a set of vertices of D such that for every two different vertices in N there is no H-restricted path in D joining ...

A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length ≤ k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N(d, k) be the smallest possible order for a k-geodetic digraph of minimum out-degree d, then N(d, k) ≥ 1 + d+ d2 + . . .+ dk = M(d, k), where M(d, k) is the Moore boun...

It is known that the spectral radius of a digraph with k edges is ≤ √ k, and that this inequality is strict except when k is a perfect square. For k = m + l, l fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph onm vertices by adding one extra vertex, and a corresponding loop, and then connecting it to the first ⌊l/2⌋ vertices by pairs of directed ed...

Partial clique covering is one of the most basic coding schemes for index coding problems, generalizing clique and cycle covering on the side information digraph and further reducing the achievable broadcast rate. In this paper, we start with partition multicast, a special case of partial clique covering with cover number 1, and show that partition multicast achieves the optimal broadcast rate ...

The line-digraph of a digraph D with vertices V , ..... V is the digrah D* obtained from D by associating with each edge of D a vertex of D*, and then directing an edge from vertex (Vi, VYj) of D* to vertex (Vk, Vm) if and only if j = k. This paper extends a characterization given by Harary and Norman for line-digraphs. It is also possible to repeatedly contract vertices of the line-digraph (wi...