From the theory of Ho0man polynomial, it is known that the adjacency matrix A of a strongly connected regular digraph of order n satis3es certain polynomial equation AP(A)=Jn, where l is a nonnegative integer, P(x) is a polynomial with rational coe5cients, and Jn is the n×n matrix of all ones. In this paper we present some su5cient conditions, in terms of the coe5cients of P(x), to ensure that ...

An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices is one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r(u) = v, Such that there are two walks of lenght ≤ k from u to v. The smallest positive integer p such that the compositio...

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that – every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) −...

We determine the minimum sum–connectivity index of bicyclic graphs with n vertices and matching number m, where 2 ≤ m ≤ ⌊n2 ⌋, the minimum and the second minimum, as well as the maximum and the second maximum sum–connectivity indices of bicyclic graphs with n ≥ 5 vertices. The extremal graphs are characterized. MSC 2000: 05C90; 05C35; 05C07

Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such...

We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we constr...

Let B(2m,m) be the set of all bicyclic graphs on 2m(m ≥ 2) vertices with perfect matchings. In this paper, we characterize the bicyclic graphs with minimal number of matchings and maximal number of independent sets in B(2m,m). 2010 Mathematics Subject Classification: 05C69, 05C05

Let G=(V,A) be a digraph. The eccentricity e(u) of a vertex u is the maximum distance from u to any other vertex in G. A vertex v in G is an eccentric vertex of u if the distance from u to v equals e(u). The eccentric digraph ED(G) of a digraph G has the same vertex set as G and has arcs from a vertex v to its eccentric vertices. In this paper we present several results on the eccentric digraph...

An upward planar drawing of a digraph G is a planar drawing of G where every edge is drawn as a simple curve monotone in the vertical direction. A digraph is upward planar if it has an embedding that admits an upward planar drawing. The problem of testing whether a digraph is upward planar is NP-complete. In this paper we give a linear-time algorithm to test the upward planarity of a series-par...

A minimum reversing set of a digraph is a smallest sized set of arcs which when reversed makes the digraph acyclic. We investigate a related issue: Given an acyclic digraph D, what is the size of a smallest tournataent T which has the arc set of D as a minimum reversing set? We show that such a T always exists and define the reversing number ofan acyclic digraph to be the number of vertices in ...