Hamiltonian-connected self-complementary graphs

AbstractA self-complementary graph having a complementing permutation σ = [1, 2, 3, …, 4k], consisting of one cycle, and having the edges (1, 2) and (1, 3) is strongly Hamiltonian iff it has an edge between two even-labelled vertices. Some of these strongly Hamiltonian self-complementary graphs are also shown to be Hamiltonian connected

Paths in r-partite self-complementary graphs

AbstractThis paper aims at finding best possible paths in r-partite self-complementary (r-p.s c.) graphs G(r). It is shown that, every connected bi-p.s.c. graphs G(2) of order p. with a bi-partite complementing permutation (bi-p.c.p) σ having mixed cycles, has a (p-3)-path and this result is best possible. Further, if the graph induced on each cycle of bi-p.c.p. of G(2) is connected then G(2) has a hamiltonian path. Lastly the fact that every r-p.s.c graph with an r-partite of σ has non-empty intersection with at least four partitions of G(r), has a hamiltonian path, is established. The graph obtained from G(r) by adding a vertex u constituting (r + 1)-st partition of G(r), which is the fixed point of σ∗ = (u)σ also has a hamiltonian path The last two results generalize the result that every self-complementary graph has a hamiltonian path

On the digraph of a unitary matrix

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.Comment: 6 page

A statistical mechanics approach to autopoietic immune networks

The aim of this work is to try to bridge over theoretical immunology and disordered statistical mechanics. Our long term hope is to contribute to the development of a quantitative theoretical immunology from which practical applications may stem. In order to make theoretical immunology appealing to the statistical physicist audience we are going to work out a research article which, from one side, may hopefully act as a benchmark for future improvements and developments, from the other side, it is written in a very pedagogical way both from a theoretical physics viewpoint as well as from the theoretical immunology one. Furthermore, we have chosen to test our model describing a wide range of features of the adaptive immune response in only a paper: this has been necessary in order to emphasize the benefit available when using disordered statistical mechanics as a tool for the investigation. However, as a consequence, each section is not at all exhaustive and would deserve deep investigation: for the sake of completeness, we restricted details in the analysis of each feature with the aim of introducing a self-consistent model.Comment: 22 pages, 14 figur

On self-complementation

We prove that, with very few exceptions, every graph of order n, n - 0, 1(mod 4) and size at most n - 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs