In this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $C(X)$ and the same of $C(Y)$, for some special subset $Y$ of $X$. For instance, if $Y=Coz(f)$ for some $fin C(X)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $C(Y)$ and the set of prime $z$-ideals of $C(X)$ not containing $f$. Moreover, c...

Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.

in this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $c(x)$ and the same of $c(y)$, for some special subset $y$ of $x$. for instance, if $y=coz(f)$ for some $fin c(x)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $c(y)$ and the set of prime $z$-ideals of $c(x)$ not containing $f$. moreover, c...

a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

‎Let $G$ be a finite group‎. ‎An element $gin G$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $G$‎, ‎$chi(g)neq 0$‎. ‎The bi-Cayley graph ${rm BCay}(G,T)$ of $G$ with respect to a subset $Tsubseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin G‎, ‎ tin T}$‎. ‎Let ${rm nv}(G)$ be the set‎ ‎of all non-vanishi...