Harmonic functions on annuli of graphs
In 1980 Bondy [2] proved that a (k+s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least ((k+1)(n+s−1)+1)/2. It is shown in [1] that one can allow exceptional (k+ 1)-sets violating this condition and still implying the considered Hamiltonian property. In this note we generalize this result for s = −1 and s = 0 and graphs that fulfill a certain connectivity condition....
Let be a simple graph. A function from the set of orientations of to the set of non-negative integers is called a continuous function on orientations of if, for any two orientations and of , whenever and differ in the orientation of exactly one edge of . We show that any continuous function on orientations of a simple graph has the interpolation property as follows: If there are two orientations and of with and , where , then for any integer such that , there are...
In this paper, we consider the intersection graph of gamma sets in the total graph on ℤₙ. We characterize the values of n for which is complete, bipartite, cycle, chordal and planar. Further, we prove that is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of .
Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group and each p which is not an even integer, there exists n ∈ ℕ and a subspace whose group of isometries is the direct product × ℤ₂.
It is shown that every 3-connected planar graph with a large number of vertices has a long induced path.
The paper studies minimal acyclic dominating sets, acyclic domination number and upper acyclic domination number in graphs having cut-vertices.