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Full-Text Articles in Physical Sciences and Mathematics
On The Zero-Divisor Graphs Of Finite Free Semilattices, Kemal Toker
On The Zero-Divisor Graphs Of Finite Free Semilattices, Kemal Toker
Turkish Journal of Mathematics
Let $SL_{X}$ be the free semilattice on a finite nonempty set $X$. There exists an undirected graph $\Gamma(SL_{X})$ associated with $SL_{X}$ whose vertices are the proper subsets of $X$, except the empty set, and two distinct vertices $A$ and $B$ of $\Gamma(SL_{X})$ are adjacent if and only if $A\cup B=X$. In this paper, the diameter, radius, girth, degree of any vertex, domination number, independence number, clique number, chromatic number, and chromatic index of $\Gamma(SL_{X})$ have been established. Moreover, we have determined when $\Gamma(SL_{X})$ is a perfect graph and when the core of $\Gamma(SL_{X})$ is a Hamiltonian graph.
On The Comaximal Ideal Graph Of A Commutative Ring, Mehrdad Azadi, Zeinab Jafari, Changiz Eslahchi
On The Comaximal Ideal Graph Of A Commutative Ring, Mehrdad Azadi, Zeinab Jafari, Changiz Eslahchi
Turkish Journal of Mathematics
Let $R$ be a commutative ring with identity. We use $\Gamma ( R )$ to denote the comaximal ideal graph. The vertices of $\Gamma ( R )$ are proper ideals of R that are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with the planarity and perfection of $\Gamma ( R )$.