Narrow Results

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W*(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ bR and b ∉ aR. Also the comaximal graph is a graph with vertices all elements of R and two distinct vertices a and b are adjacent if and only if aR + bR = R. We denote the subgraph of the comaximal graph with vertex-set W*(R), by Γ₂(R). In this note, we study the relations between two graphs Γ′(R) and Γ₂(R). Also, we investigate the cozero-divisor graphs of idealizations of commutative rings.

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, and let $f:R\to S$ be a ring homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of $R$ which are transferred to the comaximal graph of $R{\bowtie }^{f}J$, and also we study some algebraic properties of the ring $R{\bowtie }^{f}J$ by way of graph theory. The comaximal graph of $R$, $\mathrm{\Gamma }(R)$, was introduced by Sharma and Bhatwadekar in 1995. The vertices of $\mathrm{\Gamma }(R)$ are all elements of $R$ and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let ${\mathrm{\Gamma }}_2(R)$ be the subgraph of $\mathrm{\Gamma }(R)$ generated by non-unit elements, and let $J(R)$ be the Jacobson radical of $R$. It is shown that the diameter of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the diameter of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, and the girth of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the girth of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, provided some special conditions.

In this paper, two outwardly different graphs, namely, the zero-divisor graph $\mathrm{\Gamma }(C_c(X))$ and the comaximal graph ${\mathrm{\Gamma }}_2^{\prime }(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on any zero-dimensional space $X$, are investigated. It is observed that these two graphs exhibit resemblance, so far as the diameters, girths, connectedness, triangulatedness or hypertriangulatedness are concerned. However, the study reveals that the zero-divisor graph $\mathrm{\Gamma }(A_c(X))$ of an intermediate ring $A_c(X)$ of $C_c(X)$ is complemented if and only if the space of all minimal prime ideals of $A_c(X)$ is compact. Moreover, $\mathrm{\Gamma }(C_c(X))$ is complemented when and only when its subgraph $\mathrm{\Gamma }(A_c(X))$ is complemented. On the other hand, the comaximal graph of $C_c(X)$ is complemented if and only if the comaximal graph of its over-ring $C(X)$ is complemented and the latter graph is known to be complemented if and only if $X$ is a $P$-space. Indeed, for a large class of spaces (i.e. for perfectly normal, strongly zero-dimensional spaces which are not P-spaces), $\mathrm{\Gamma }(C_c(X))$ and ${\mathrm{\Gamma }}_2^{\prime }(C_c(X))$ are seen to be non-isomorphic. Defining appropriately the quotient of a graph, it is utilized to establish that for a discrete space $X$, $\mathrm{\Gamma }(C_c(X))$ (= $\mathrm{\Gamma }(C(X))$) and ${\mathrm{\Gamma }}_2^{\prime }(C_c(X))$ (= ${\mathrm{\Gamma }}_2^{\prime }(C(X))$) are isomorphic, if $X$ is at most countable. Under the assumption of continuum hypothesis, the converse of this result is also shown to be true.

For a ring $R$ (not necessarily commutative) with identity, the comaximal graph of $R$, denoted by $\mathrm{\Omega }(R)$, is a graph whose vertices are all the nonunit elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph ${\mathrm{\Omega }}_1(R)$ of $\mathrm{\Omega }(R)$ induced by $R\backslash U_{\ell }(R)$, where $U_{\ell }(R)$ is the set of all left-invertible elements of $R$. We characterize those rings $R$ for which ${\mathrm{\Omega }}_1(R)\backslash J(R)$ is a complete graph or a star graph, where $J(R)$ is the Jacobson radical of $R$. We investigate the clique number and the chromatic number of the graph ${\mathrm{\Omega }}_1(R)\backslash J(R)$, and we prove that if every left ideal of $R$ is symmetric, then this graph is connected and its diameter is at most 3. Moreover, we completely characterize the diameter of ${\mathrm{\Omega }}_1(R)\backslash J(R)$. We also investigate the properties of $R$ when ${\mathrm{\Omega }}_1(R)$ is a split graph.

Let $R$ be a commutative ring with nonzero identity. The comaximal graph of $R$, denoted by $\mathrm{\Gamma }(R)$, is a simple graph with vertex set $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let ${\mathrm{\Gamma }}_2(R)$ be an induced subgraph of $\mathrm{\Gamma }(R)$ with nonunit elements of $R$ as vertices. In this paper, we describe the normalized Laplacian spectrum of $\mathrm{\Gamma }({\mathbb{Z}}_n)$, and we determine it for some values of $n$, where ${\mathbb{Z}}_n$ is the ring of integers modulo $n$. Moreover, we investigate the normalized Laplacian energy and general Randic index of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$.

In this paper, we investigate the projectivity of the comaximal graph of a finite lattice.

Let R be a commutative ring with unity, and let $\mathrm{\Gamma }(R)$ denote the comaximal graph of R. The comaximal graph $\mathrm{\Gamma }(R)$ has vertex set as R, and any two distinct vertices x, y of $\mathrm{\Gamma }(R)$ are adjacent if $Rx+Ry=R$. Let ${\mathrm{\Gamma }}_2(R)$ denote the induced subgraph of $\mathrm{\Gamma }(R)$ on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of ${\mathrm{\Gamma }}_2(R)$ are adjacent if $Rx+Ry=R$. In this paper, we study the graphical structure as well the adjacency spectrum of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, where $n\ge 4$ is a non-prime positive integer, and ${\mathbb{Z}}_n$ is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ are $0$ with multiplicity $n-\phi (n)-D-1$, and remaining eigenvalues are contained in the spectrum of a symmetric $D\times D$ matrix. We further calculate the rank and nullity of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$. We also determine all the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ whenever ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ is a bipartite graph. Finally, apart from determining certain structural properties of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, we conclude the paper by determining the metric dimension of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$.