Narrow Results

The intersection non-simple graph, denoted by $\mathrm{\text{INS}}(G)$, of a finite abelian group $G$ is an undirected graph whose vertex set is the collection of all proper non-trivial subgroups of $G$, and any two distinct vertices are adjacent if and only if their intersection is not a simple subgroup of $G$. We obtain some properties of $\mathrm{\text{INS}}(G)$ related to connectedness, completeness, degree, and girth. The concepts of bipartiteness, triangle-free, cluster, claw-free and cograph are taken into consideration. We also investigate the clique number, independence number, domination number, and planarity of $\mathrm{\text{INS}}(G)$.

We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

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 V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for . For instance, we find the domination number of . We also find the chromatic number of in some cases. Furthermore, we study cycles in , and complete subgraphs in determining the structure of V for which is planar.

To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

Let $R$ be a commutative ring with identity and $I^{\ast }(R),$ the set of all nontrivial proper ideals of $R$. The intersection graph of ideals of $R$, denoted by ${ℐ}(R)$, is a simple undirected graph with vertex set as the set $I^{\ast }(R)$, and, for any two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J\ne (0)$. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring $R$, ${ℐ}(R)$ is a projective graph if and only if $R\cong R_1\times F_1,$ where $R_1$ is a local principal ideal ring with maximal ideal ${𝔪}_1$ of nilpotency three and $F_1$ is a field. Furthermore, it is shown that for an Artinian ring $R,$$\overline{\gamma }({ℐ}(R))=2$ if and only if $R\cong R_1\times R_2,$ where each $R_i(1\le i\le 2)$ is a local principal ideal ring with maximal ideal ${𝔪}_i\ne (0)$ such that ${𝔪}_i^2=(0).$

Let $F$ be a field and ${\mathrm{\text{GL}}}_n(F)$ the general linear group of degree $n>1$ over $F$. The intersection graph $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ of ${\mathrm{\text{GL}}}_n(F)$ is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of ${\mathrm{\text{GL}}}_n(F)$. Two vertices $A$ and $B$ of $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ are adjacent if $A\ne B$ and $A\cap B\ne \{I_n\}$. In this paper, we show that if $F$ is a finite field containing at least three elements, then the diameter $\delta (\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F)))$ is $2$ or $3$. We also classify ${\mathrm{\text{GL}}}_n(F)$ according to $\delta (\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F)))$. In case $F$ is infinite, we prove that $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ is one-ended of diameter $2$ and its unique end is thick.

Consider a graph D and a family FI of connected edge subgraphs of D. Let GI(V, F) be the intersection graph of FI and G the overlap graph of FI. We describe polynomial time algorithms for subgraph overlap graphs G when their intersection graphs GI have specific hereditary properties. The algorithms are to find maximum induced complete bipartite subgraphs, maximum weight holes of a given parity, minimum dominating holes, antiholes of a given parity and some others. In addition, we define the family of subgraph filament graphs based on D, FI and GI, and prove it to be the same as the family of subgraph overlap graphs.

Let (P, ≤) be a partially ordered set (poset, briefly) with a least element 0. The intersection graph of ideal of P, denoted by G(P), is a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ {0}. In this paper, we study some relations between the algebraic properties of posets and graph-theoretic properties of G(P). We investigate the connectivity, diameter, girth and planarity of the intersection graph. Also, among the other things, we show that if the clique number of G(P) is finite, then P is finite too.

Let R be a commutative ring. The intersection graph of gamma sets in the zero-divisor graph Γ(R) of R is the graph I_Γ(R) with vertex set as the collection of all gamma sets of the zero-divisor graph Γ(R) of R and two distinct vertices A and B are adjacent if and only if A ∩ B ≠ ∅. In this paper, we study about various properties of I_Γ(R) and investigate the interplay between the graph theoretic properties of I_Γ(R) and the ring theoretic properties of R.

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I{(R)}^{\ast }$ be the set of all nontrivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I{(R)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\ne \{0\}$. For every multiplication $R$-module $M$, the diameter and the girth of $G_M(R)$ are determined. Among other results, we prove that if $M$ is a faithful $R$-module and the clique number of $G_M(R)$ is finite, then $R$ is a semilocal ring. We denote the ${\mathbb{Z}}_n$-intersection graph of ideals of the ring ${\mathbb{Z}}_m$ by $G_n({\mathbb{Z}}_m)$, where $n,m\ge 2$ are integers and ${\mathbb{Z}}_n$ is a ${\mathbb{Z}}_m$-module. We determine the values of $n$ and $m$ for which $G_n({\mathbb{Z}}_m)$ is perfect. Furthermore, we derive a sufficient condition for $G_n({\mathbb{Z}}_m)$ to be weakly perfect.

Let $m>1$ be an integer, and let $I{({\mathbb{Z}}_m)}^{\ast }$ be the set of all non-zero proper ideals of ${\mathbb{Z}}_m$. The intersection graph of ideals of ${\mathbb{Z}}_m$, denoted by $G({\mathbb{Z}}_m)$, is a graph with the vertex set $I{({\mathbb{Z}}_m)}^{\ast }$ and two distinct vertices $I,J\in I{({\mathbb{Z}}_m)}^{\ast }$ are adjacent if and only if $I\cap J\ne 0$. Let $n>1$ be an integer and ${\mathbb{Z}}_n$ be a ${\mathbb{Z}}_m$-module. In this paper, we study a kind of graph structure of ${\mathbb{Z}}_m$, denoted by $G_n({\mathbb{Z}}_m)$. It is the undirected graph with the vertex set $I{({\mathbb{Z}}_m)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $I{\mathbb{Z}}_n\cap J{\mathbb{Z}}_n\ne 0$. Clearly, $G_m({\mathbb{Z}}_m)=G({\mathbb{Z}}_m)$. Let $m=p_1^{{\alpha }_1}\cdots p_s^{{\alpha }_s}$ and $n=p_1^{{\beta }_1}\cdots p_s^{{\beta }_s}$, where $p_i$’s are distinct primes, ${\alpha }_i$’s are positive integers, ${\beta }_i$’s are non-negative integers, and $0\le {\beta }_i\le {\alpha }_i$ for $i=1,\dots ,s$ and let $S=\{1,\dots ,s\}$, $S^{\prime }=\{i\in S:{\beta }_i\ne 0\}$. The cardinality of $S^{\prime }$ is denoted by $s^{\prime }$. Also, let $\alpha (G_n({\mathbb{Z}}_m))$, $\gamma (G_n({\mathbb{Z}}_m))$ and $𝔄$ denote the independence number, the domination number and the set of all isolated vertices of $G_n({\mathbb{Z}}_m)$, respectively. We prove that $gr(G_n({\mathbb{Z}}_m))\in \{3,\infty \}$ and we show that if $G_n({\mathbb{Z}}_m)$ is not a null graph, then

The present paper is a review of the current state of Graph-Link Theory (graph-links are closely related to homotopy classes of looped interlacement graphs): a theory suggested in,^1,2 see also,³ dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. In this paper we show how some methods of classical and virtual knot theory can be translated into the language of abstract graphs, and some theorems can be reproved and generalised to this graphical setting. We construct various invariants, prove certain minimality theorems and construct functorial mappings for graph-knots and graph-links. In this paper, we first show non-equivalence of some graph-links to virtual links.