Let $$F_1$$ and $$F_2$$ be two disjoint graphs. The union $$F_1\cup F_2$$ is a graph with vertex set $$V(F_1)\cup V(F_2)$$ edge $$E(F_1)\cup E(F_2)$$ , the join $$F_1+F_2$$ E(F_2)\cup \{xy\;|\; x\in V(F_1)\hbox { } y\in V(F_2)\}$$ . In this paper, we present characterization to $$(P_5, K_1\cup K_3)$$ -free graphs, prove that $$\chi (G)\le 2\omega (G)-1$$ if G -free. Based on result, further $$ ...

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...