A semiprime filter-based identity- summand graph of a lattice

Abstract

Let $F$ be a proper filter of a lattice $L$ with the least
element $0$ and the greatest element $1$. The filter-based
identity-summand graph of $L$ with respect to $F$, denoted by
$\Gamma_{F} (L)$, is the graph with vertices $I^*_{F} (L) = \{x
\in L \setminus F: x \vee y \in F \, \, \mbox{for some} \, \, y
\in L \setminus F \}$, and distinct vertices $x$ and $y$ are
adjacent if and only if $x \vee y \in F$. We will make an
intensive study of the notions of diameter, grith, chromatic
number, clique number, independence number, domination number
and planar property of this graph. Moreover, Beck$^s
conjecture is proved for $\Gamma_{F} (L)$.