Nonstandard Graphs, Revised

This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction on an internal set from a sequence of subsets of the real line, but now the individual entities are the vertices of the graphs instead of real numbers. The transfer principle is then invoked to extend several graph-theoretic results to the nonstandard case. After incidences and adjacencies between nonstandard vertices and edges are defined, several formulas regarding numbers of vertices and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for these nonstandard graphs. Key Words: Nonstandard graphs, transfer principle, ultrapower constructions

Nonstandard Transfinite Graphs and Networks of Higher Ranks

In Chapter 8 of the Book, ``Graphs and Networks: Transfinite and Nonstandard'' (published by Birkhauser-Boston in 2004), nonstandard versions of transfinite graphs and of electrical networks having such graphs were defined and examined but only for the first two ranks, 0 and 1, of transfiniteness. In the present work, these results are extended to higher ranks of transfinteness. Such is done in detail for the natural-number ranks and also for the first transfinite ordinal rank. Results for still higher ranks of transfiniteness can be established in much the same way. Once the transfinite graphs of higher ranks are established, theorems concerning the existence of hyperreal operating points and the satisfaction of Kirchhoff's laws in nonstandard networks of higher ranks can be proven just as they are for nonstandard networks of the first rank.Comment: 8 pages, 0 figure

Ordinal Distances in Transfinite Graphs

An ordinal-valued metric taking its values in the set of all countable ordinals can be assigned to a metrizable set of nodes in a transfinite graph. Then, a variety of results concerning nodal eccentricities, radii, diameters, centers, peripheries, and blocks can be extended to transfinite graphs

A Shorter Proof of the Transitivity of Transfinite Connectedness

A criterion is established for the transitivity of connectedness in a transfinite graph. Its proof is much shorter than a prior argument published previously for that criterion.Comment: 6 page

Nonstandard Graphs

From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the vertices of the graphs instead of real numbers. The transfer principle is then invoked to extend several graph-theoretic results to the nonstandard case. After incidences and adjacencies between nonstandard vertices are defined, several formulas regarding numbers of vertices and edges, and nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for these nonstandard graphs

Hyperreal Waves on Transfinte, Terminated, Distortionless and Lossless, Transmission Lines

A prior work (see Chapter 8 of the book, ``Graphs and Networks: Transfinite and Nonstandard,'' Birkhauser-Boston, Cambridge, Mass., USA, 2004) examined the propagation of an electromagnetic wave on a transfinite transmission line, transfinite in the sense that infinitely many one-way infinite transmission lines are connected in cascade. That there are infinitely many such one-way infinite lines results in the wave propagating without ever reflecting at some discontinuity. The present work examines the cascade where the cascade terminates after only finitely many one-way infinite transmission lines, with the result that reflected waves are now produced at both the far end as well as at the initial end of the transfinite transmission line. The question of whether the reflected waves are infini tesimal or appreciable and whether they sum to an infinitesimal or appreciable amount are resolved for both distortionless and lossless lines. Finally, the generalizations to higher ranks of transfiniteness is briefly summarized.Comment: 16 pages, 3 figure

The Galaxies of Nonstandard Enlargements of Infinite and Transfinite Graphs

The galaxies of nonstandard enlargements of conventionally infinite as well as of transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their "closeness" to the principal galaxy. If an enlargement has a galaxy different from its principal galaxy, then it has a two-way infinite sequence of galaxies that are totally ordered according to that "closeness" property. There may be many such totally ordered sequences.Comment: 24 pages, 3 figure

Nonstandard Digraphs

Nonstandard graphs have been defined and examined in prior works. The present work does the same for nonstandard digraphs. Since digraphs have more structure than do graphs, the present discussion requires more complicated definitions and yields a variety of results peculiar to nonstandard digraphs. A nonstandard digraph can be obtained by means of an ultrapower construction based on a sequence of digraphs or more elegantly by using the transfer principle. We use either or both techniques in particular circumstances. As special cases, we have the enlargement of a single infinite digraph and also hyperfinite digraphs based on sequences of finite digraphs. Also examined are such ideas as incidences and adjacencies for nonstandard arcs and vertices, connectedness, components, and galaxies in nonstandard digraphs.Comment: 15 page

The Galaxies of Nonstandard Enlargements of Transfinite Graphs of Higher Rsnks

In a prior work, the galaxies of the nonstandard enlargements of conventionally infinite graphs and also of transfinite graphs of the first rank of transfiniteness were defined, examined, and illustrated by some examples. In this work it is shown how the results of the prior work extend to graphs of higher ranks.Comment: 12 page

A Circuit-Theoretic Anomaly Resolved by Nonstandard Analysis

An anomaly in electrical circuit theory is the disappearance of some of the energy when two capacitors, one charged and the other uncharged, are connected together through resistanceless wires. Nonstandard analysis shows that, when the wires are taken to have infinitesimally small but nonzero resistance, the energy dissipated in the wires equals that substantial amount of enregy that had disappeared, and that all but an infinitesimal amount of this dissipation occurs during an infinitesimal initial time period.Comment: 8 pages, 2 figure