‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

Callery pear ( Pyrus calleryana ) is an invasive tree across much of the eastern United States that can form dense thickets, and branches stems are often covered in sharp thorns. Landowners land managers attempting to manage callery infestations faced with challenge killing and/or removing trees while also avoiding thorn damage equipment, which lead wasted time increased costs. We evaluated fir...

Homobasidiomycetes include approximately 13,000 described species of mushroom-forming fungi and related taxa. The higher-level classification of this ecologically important group has been unsettled for over 100 years. The goals of the present study were to evaluate a recent phylogenetic classification by Hibbett and Thorn that divided the homobasidiomycetes into eight major unranked clades, and...

commutes up to homotopy where T is the map that exchanges factors. Let L be a space and let be a fibration over L classified by a mapf L BF (the classifying space of stable spherical fibrations). We can form the Thom spectrum T(f) of f as a suspension spectrum by letting (T(f)) be the Thorn complex of L n--YBFn where L is the n-skeleton of L. This makes T(f)-{(T(f))n) into a suspension spectrum...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

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in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.