Volume 15, 2009, Number 4

Data in the social sciences can often modeled using signed graphs, graphs where every edge has a sign + or −, or marked graphs, graphs where every vertex has a sign + or −. The path graph P_k(G) of a graph G is obtained by representing the paths P_k in G by vertices whenever the corresponding paths P_k in G form a path P_k+1 or a cycle C_k. In this note, we introduce a natural extension of the notion of path graphs to the realm of signed graphs. It is shown that for any signed graph S, P_k(S) is balanced. The concept of a line signed graph is generalized to that of a path signed graphs. Further, in this note we discuss the structural characterization of path signed graphs. Also, we characterize signed graphs which are switching equivalent to their path signed graphs P₃(S) (P₄(S)).

Let A, B and C be rhotrices. In this paper we investigate systems of the form AB = C and come up with conditions necessary for their solvability. We also outline a direct procedure for computing an n-th root of a rhotrix.

Certain characteristics of Pythagorean triples are analysed using Integer Structure via the modular ring, Z₄. The fact (as shown by Fermat’s Last Theorem) that all the components of a triple cannot simultaneously be an even power n (with ½n even) is illustrated via the spectra of the right-end-digits of the components.

In a series of papers the author studied some properties of the well-known arithmetic functions φ and σ (see, e.g. [5, 7]. With respect to these research, a new function was defined in [1, 2] and some of its properties were discussed there. Here we shall continue this research.