Maximum Stirling numbers of the second kind.
The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.
For a connected graph of order and an ordering , of the vertices of , , where is the distance between and . The traceable number of is defined by where the minimum is taken over all sequences of the elements of . It is shown that if is a nontrivial connected graph of order such that is the length of a longest path in and is the maximum size of a spanning linear forest in , then and both these bounds are sharp. We establish a formula for the traceable number of...
We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings...
We prove that, for any prime , there are precisely medial quasigroups of order , up to isomorphism.
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called -quasigroups. We show that an incidence structure associated with a medial quasigroup of type , , is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order and dimension , or with a desarguesian affine plane of order then there is a medial quasigroup of...
Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.
For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is , the vertex-to-edge distance sum d₁(v) of v is , the edge-to-vertex distance sum d₂(e) of e is and the edge-to-edge distance sum d₃(e) of e is . The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set M₁(G) of all vertices v for which d₁(v) is minimum is the vertex-to-edge median of G; the set M₂(G) of all edges e for which d₂(e) is minimum is the edge-to-vertex median...
Two numerical invariants and of a graph, related to the concept of median, are studied.