[2] **viXra:1211.0081 [pdf]**
*submitted on 2012-11-14 10:59:09*

**Authors:** Natasha Lee, Joan Portmann

**Comments:** 5 Pages.

The paper deals with the problems of characterization of simple graphical partitions belonging to the solid graphs, i.e. graphs, in which there are no four of vertices such that it is possible some shift of edges incidental to them and with characterization of the one class of steady graphs too. The necessary and sufficient
conditions for the partition belonging to the solid graph have been established.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1211.0074 [pdf]**
*submitted on 2012-11-13 09:11:35*

**Authors:** Linfan Mao

**Comments:** 46 Pages.

Different from the system in classical mathematics, a
Smarandache system is a contradictory system in which an axiom behaves in at least
two different ways within the same system, i.e., validated and invalided, or
only invalided but in multiple distinct ways. Such systems exist extensively
in the world, particularly, in our daily life. In this paper, we discuss such a
kind of Smarandache system, i.e., non-solvable ordinary differential equation
systems by a combinatorial approach, classify these systems and characterize
their behaviors, particularly, the sum-stability and prod-stability of such
linear and non-linear differential equations. Some applications of such systems
to other sciences, such as those of globally controlling of infectious diseases,
establishing dynamical equations of instable structure, particularly, the
n-body problem and understanding global stability of matters with multilateral
properties can be also found.

**Category:** Combinatorics and Graph Theory