Let R be a commutative ring and M be an R-module. The M-intersection graph of ideals of R, denoted by GM(R) is a graph with the vertex set I(R) ∗, , and two distinct vertices I and J are adjacent if and only if IM ∩,JM ̸, = 0. In this paper, we study GR/J (R/I), where I and J are ideals of R and I ⊆,J. We characterize all ideals I and J for which GR/J (R/I) is planar, outerplanar or ring graph.

The crossing number of a graph G is the minimum number of edge crossings over all possibl e drawings of G in a plane. The crossing number is an important measure of the non-planarity of a graph, with ap plications in discrete and computational geometry and VLSI circuit design. In this paper we introduce vertex cente red crossing number and study the same for maximal planar graph.

The main aim of this paper is to extend the recently developed methods for calculating the buckling loads of planar symmetric frames to include the effect of semi-rigidity of the joints. This is achieved by decomposing a symmetric model into two submodels and then healing them in such a manner that the union of the eigenvalues of the healed submodels result in the eigenvalues of the entire model. Thus the critical load of the frame is obtained using the eigenvalues of its submodels.

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Let R be a commutative ring with identity. We use j (R) to denote the comaximal ideal graph. The vertices of j (R) are proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I+J=R. In this paper we show some properties of this graph together with planarity of line graph associated to j (R).

This paper is dedicated to propose an algorithm in order to generate the certain isomers of some well-known fullerene bases. Furthermore, we enlist the bipartite edge frustration correlated with some of symmetrically distinct infinite families of fullerenes generated by the offered process.

The complement of the intersection graph of subgroups of a group $G$, denoted by $\mathcal{I}^c(G)$, is the graph whose vertex set is the set of all nontrivial proper subgroups of $G$ and its two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K$ is trivial. In this paper, we classify all finite groups whose complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or $C_3$-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.