Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-module M , 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.

In this paper some basic mathematical properties for the third and hyper Zagreb coindices of graph operations containing the Cartesian product and composition will be explained.

Connected cubic graphs of order 10p 3 which admit an automorphism group acting semisymmetrically are investigated. We prove that every connected cubic edge-transitive graph of order 10p 3 is vertex-transitive, where p is a prime.

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Abstract:In this research study, several topological indices have been investigated for linear [n]-Tetracene, V-Tetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori. The calculated indices are first, second, third and modified second Zagreb indices. In addition, the first and second Zagreb coindices of these nanostructures were calculated. The explicit formulae for connectivity indices of various families of Tetracenic nanotubes and nanotori are presented in this manuscript. These formulae correlate the chemical structure of nanostructures to the information about their physical features.