let $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

A proof assistant is interactive computer software that humans use to prepare scripts of mathematical proofs. These proof scripts can be parsed and verified directly from the fundamental rules of logic and the foundational axioms of mathematics. The technology underlying proof assistants and formal proofs has been under development for decades and grew out of efforts in the early twentieth cent...

This paper presents a new mathematical model for a bi-objective job shop scheduling problem with sequence-dependent setup times that minimizes the weighted mean completion time and the weighted mean tardiness time. For solving this multi-objective model, we develop a fuzzy multi-objective linear programming (FMOLP) model. In this problem, a proposed FMOLP method is applied with respect to the o...

we show that in {bf zf} set theory without choice, the ultrafilter mbox{principle} ({bf up}) is equivalent to several compactness theorems for alexandroff discrete spaces and to rudin's lemma, a basic tool in topology and the theory of quasi-continuous domains. important consequences of rudin's lemma are various lift lemmas, saying that certain properties of posets are inherited by th...