0. Initiated by Sekanina [6], in the sixties and more intensively in the seventies there were considered powers of undirected graphs with special respect to their Hamiltonian behaviour. These investigations have resulted in a lot of interesting and partly very profound propositions; we only remind of the (simple) result of Sekanina [6] that the cube of every finite connected graph is Hamiltonia...

Let G be a digraph. For two vertices u and v in G, the distance d(u, v) from u to v in G is the length of the shortest directed path from u to v. The eccentricity e(v) of v is the maximum distance of v to any other vertex of G. A vertex u is an eccentric vertex of v if the distance from v to u is equal to the eccentricity of v. The eccentric digraph ED(G) of G is the digraph that has the same v...

It is easily shown that every digraph with m edges has a directed cut of size at least m/4, and that 1/4 cannot be replaced by any larger constant. We investigate the size of a largest directed cut in acyclic digraphs, and prove a number of related results concerning cuts in digraphs and acyclic digraphs.

We consider a hierarchy of four classes of interval digraphs, or equivalently four classes of 0,1-matrics. We provide forbidden submatrix characterizations separating the successive classes. The largest class is the set of (adjacency matrices of) interval digraphs; the smallest is the set of (adjacency matrices of) unit interval digraphs.

We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called ‘crowns’. Moreover, there is no homomorphism from any of our digraphs onto Z. The methods are adapted from [5] and [6].

For given two digraphs, we can construct a larger digraph through join. The digraphs that make up the join are called factors of In this paper, give necessary and sufficient condition function on determined by discrete Morse functions is function. Moreover, further prove theory when satisfy certain conditions.

Abstract Associative spectra of graph algebras are examined with the help homomorphisms DFS trees. Undirected graphs classified according to associative their algebras; there only three distinct possibilities: constant 1, powers 2, and Catalan numbers. antiassociative digraphs described, determined for certain families digraphs, such as paths, cycles, on two vertices.

The digraphs P (n, k) have vertices corresponding to length k permutations of an n set and arcs corresponding to (k + 1) permutations. Answering a question of Starling, Klerlein, Kier and Carr we show that these digraphs are Hamiltonian for k ≤ n − 3. We do this using restricted Eulerian cycles and the fact that P (n, k) is nearly the line digraph of P (n, k−1). We also show that the digraphs P...

We introduce adequate concepts of expansion of a digraph to obtain a sequential construction of minimal strong digraphs. We obtain a characterization of the class of minimal strong digraphs whose expansion preserves the property of minimality. We prove that every minimal strong digraph of order n > 2 is the expansion of a minimal strong digraph of order n — \ and we give sequentially generative...

The conventional binary operations of cartesian product, conjunction, and composition of two digraphs D, and D2 are observed to give the sum, the product, md a more complicated combination of the spectra of D1 and Dz as the resulting spectrum. These formulas for analyzing the spectrum of a digraph are utilized to construct for any positive integer PI, a collection of n nonisomorphic strung regu...