Compositions, Generated by Special Nets in Affinely Connected Spaces

Special nets which characterize Cartesian, geodesic, Chebyshevian, geodesic- Chebyshevian and Chebyshevian-geodesic compositions are introduced. Con- ditions for the coefficients of the connectedness in the parameters of these special nets are found.

On Compositions in Equiaffine Space

In an equiaffine space q N E using the connection define with projective tensors na and ma the connections 1 , 2 and 3 . For the spaces N N 1A ,2A and N 3A , with coefficient of connection 1 , 2 and 3 respectively, we proved that the affinor of composition and the projective affinors have equal covariant derivatives. It follows that the connection 3 is equaffine as well, and the connections and 3 are projective to each other. In the case where q N E and N 3A have equal Ricci tensors, we find the fundamental nvector . In [4] compositions with structural affinor a are studied. Space containing compositions with symmetric connection and Weyl connection are studied in [6] and [7] respectively.

Conjugate Compositions in Even-Dimensional Affinely Connected Spaces without a Torsion

Let in even-dimensional a±nely connected space without a torsion A2m be given a composition Xm£Xm by the affinor a¯ ®. The affinor b¯ ®, determined with the help of the eigen-vectors of the matrix (a¯ ®), de¯nes the second composition Ym £ Y m. Conjugate compositions are introduced by the condition: the a±nors of any of both compositions transform the vectors from the one position of the composition, generated by the other a±nor, in the vectors from the another its position. It is proved that the compositions de¯ne by a±nors a¯ ® and b¯ ® are conjugate. It is proved also that if the composition Xm£Xm is Cartesian and composition Ym£Y m is Cartesian or chebyshevian, or geodesic than the space A2m is affine.

On Compositions in Equiaffine Space

In an equiaffine space q N E using the connection define with projective tensors na and ma the connections 1 , 2 and 3 . For the spaces N N 1A ,2A and N 3A , with coefficient of connection 1 , 2 and 3 respectively, we proved that the affinor of composition and the projective affinors have equal covariant derivatives. It follows that the connection 3 is equaffine as well, and the connections and 3 are projective to each other. In the case where q N E and N 3A have equal Ricci tensors, we find the fundamental nvector . In [4] compositions with structural affinor a are studied. Space containing compositions with symmetric connection and Weyl connection are studied in [6] and [7] respectively.

The Eccentric Connectivity Polynomial of some Graph Operations

The eccentric connectivity index of a graph G, ξ^C, was proposed by Sharma, Goswami and Madan. It is defined as ξ^C(G) = ∑ u ∈ V(G) degG(u)εG(u), where degG(u) denotes the degree of the vertex x in G and εG(u) = Max{d(u, x) | x ∈ V (G)}. The eccentric connectivity polynomial is a polynomial version of this topological index. In this paper, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented.