By left magma-$e$-magma, I mean a set containingthe fixed element $e$, and equipped by two binary operations "$cdot$", $odot$ with the property $eodot (xcdot y)=eodot(xodot y)$, namelyleft $e$-join law. So, $(X,cdot,e,odot)$ is a left magma-$e$-magmaif and only if $(X,cdot)$, $(X,odot)$ are magmas (groupoids), $ein X$ and the left $e$-join law holds.Right (and two-sided) magma-$e$-magmas are de...

A (p,q) graph G is called super edge-magic if there exists a bijective function f from V (G) ∪ E(G) to {1, 2,. .. , p + q} such that f (x) + f (xy) + f (y) is a constant k for every edge xy of G and f (V (G)) = {1, 2,. .. , p}. Furthermore, the super edge-magic deficiency of a graph G is either the minimum nonnegative integer n such that G ∪ nK 1 is super edge-magic or +∞ if there exists no suc...

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214—217), proved that the characteristic polynomial xW of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. // x is a modular flat of G, x(G)/x(x) = X(7^(G))/(\ 1), where TX{G) is the complete Brown truncation of G by x. (The...

The study of graphs and networks accomplished by topological measures plays an applicable task to obtain their hidden topologies. This procedure has been greatly used in cheminformatics, bioinformatics, biomedicine, where estimations based on graph invariants have made available for effectively communicating with the different challenging tasks. Irregularity are mostly characterization nonregul...

A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, torus, projective plane etc. The purpose of this lecture is to explore some of these topological a...