Narrow Results

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT_H(R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT_H(R).

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z₀(Γ(R)) and T₀(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z₀(Γ(R)) and T₀(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T₀(Γ(R)).

In this paper, we initiate the study of the total zero-divisor graph over a commutative ring with unity. This graph is constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring and give a joint insight of the structure of zero-divisors in a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graph of the ring ${\mathbb{Z}}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb{Z}}_m$ and ${\mathbb{Z}}_n$ are isomorphic if and only if $m=n$.

In this paper, we investigate the upper and lower bounds for the sum of domination number of a graph and its total graph and characterize the extremal graphs.

Communication networks must be constructed to be as stable as possible, not only with the respect to the initial disruption, but also with respect to the possible reconstruction. Many graph theoretical parameters have been used to describe the stability of communication networks. Tenacity is a reasonable one, which shows not only the difficulty to break down the network but also the damage that has been caused. Total graphs are the largest graphs formed by the adjacent relations of elements of a graph. Thus, total graphs are highly recommended for the design of interconnection networks. In this paper, we determine the tenacity of the total graph of a path, cycle and complete bipartite graph, and thus give a lower bound of the tenacity for the total graph of a graph.

A signed graph is an ordered pair $S=(S^u,\sigma )$, where $S^u$ is a graph G = (V, E), called the underlying graph of S and $\sigma :E\to \{+,-\}$ is a function from the edge set E of S^u into the set {+, -}, called the signature of S. In this paper, we characterize all those signed graphs whose 2-path signed graphs are isomorphic to their square signed graph along with algorithm to check the same. In other sections we find the characterization of signed graph S such that $D_2\cong D^2$ where D is a derived signed graph of the signed graph S such as: line signed graphs, total signed graphs, common edge signed graphs, splitting signed graphs. Also each characterization is supported by algorithms for the same.

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by Γ_S(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of Γ_S(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by T_Γ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of T_Γ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in T_Γ(R) is denoted by I_TΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph I_TΓ (ℤ_n) of gamma sets in T_Γ(ℤ_n). In this paper, we study about I_TΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of I_TΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam(I_TΓ(R)) ≤ 2 and gr(I_TΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of I_TΓ(R) are given. Further, we discuss about the vertex-transitive property of I_TΓ(R). At last, we obtain all commutative Artin rings R for which I_TΓ(R) is either planar or toroidal or genus two.

The intersection graph I_TΓ(R) of gamma sets in the total graph T_Γ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about I_TΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on I_TΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of I_TΓ(R). Further, we focus on certain graph theoretic parameters of I_TΓ(R) like the independence number, the clique number and the connectivity of I_TΓ(R). Also, we obtain both vertex and edge chromatic numbers of I_TΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(I_TΓ(R)) = ω(I_TΓ(R)). Having proved that I_TΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which I_TΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which I_TΓ(R) is of class one (i.e. χ′(I_TΓ(R)) = Δ(I_TΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of I_TΓ(R) are equal to the degree of any vertex in I_TΓ(R).

Let $F$ be a finite field, $R=T_n(F)$ the ring of all $n\times n$ upper triangular matrices over $F$, $Z(R)$ the set of all zero-divisors of $R$, i.e. $Z(R)$ consists of all $n\times n$ upper triangular singular matrices over $F$. The total graph of $R=T_n(F)$, denoted by ${𝒯}(\mathrm{\Gamma }(R))$, is a graph with all elements of $R$ as vertices, and two distinct vertices $x,y\in R$ are adjacent if and only if $x+y\in Z(R)$. In this paper, we determine all automorphisms of the total graph ${𝒯}(\mathrm{\Gamma }(R))$ of $R=T_n(F)$.

We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors. We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.