**Volume 15** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Note on path signed graphs**
*Original research paper. Pages 1—6*
P. Siva Kota Reddy and M. S. Subramanya

Full paper (PDF, 129 Kb) |

Abstract 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 P_{3}(S) (P_{4}(S)).

** On the linear systems over rhotrices**

*Original research paper. Pages 7—12*

Abdulhadi Aminu

Full paper (PDF, 147 Kb) | Abstract

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.

**Structure and spectra of the components of primitive Pythagorean triples and Fermat’s last theorem**

*Original research paper. Pages 13—22*

J. V. Leyendekkers, A. G. Shannon and C. K. Wong

Full paper (PDF, 204 Kb) | Abstract

Certain characteristics of Pythagorean triples are analysed using Integer Structure via the modular ring, *Z*_{4}. The fact (as shown by Fermat’s Last Theorem) that all the components of a triple cannot simultaneously be an even power *n* (with ½*n* even) is illustrated via the spectra of the right-end-digits of the components.

**A remark on an arithmetic function. Part 3**

*Original research paper. Pages 23—27*

Krassimir Atanassov

Full paper (PDF, 126 Kb) | Abstract

In a series of papers the author studied some properties of the well-known arithmetic functions

*φ* and

*σ* (see, e.g.

). With respect to these research, a new function was defined in

and some of its properties were discussed there. Here we shall continue this research.

**Volume 15** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4