A circulant is a Cayley graph of a cyclic group Arc transitive circulants of square free order are classi ed It is shown that an arc transitive circulant of square free order n is one of the following the lexicographic product Kb or the deleted lexicographic Kb b where n bm and is an arc transitive circulant or is a normal circulant that is Aut has a normal regular cyclic subgroup

A construction of minimum cycle bases of the lexicographic product of graphs is presented. Moreover, the length of a longest cycle of a minimal cycle basis is determined.

The forgotten topological index or F-index of a graph is defined as the sum of cubes of the degree of all the vertices of the graph. In this paper we study the F-index of four operations related to the lexicographic product on graphs which were introduced by Sarala et al. [D. Sarala, H. Deng, S.K. Ayyaswamya and S. Balachandrana, The Zagreb indices of graphs based on four new operations related...

Let γb(G) denote the broadcast domination number for a graph G. In [Discrete Applied Math. 154 (2006), 59–75], Dunbar et al. determined the value of γb(G), where G is the Cartesian product of two paths. In this paper, we evaluate the value of γb(G), whenever G is the strong product, the direct product and the lexicographic product of two paths.

The standard products—cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by W. Imrich and H. Izbicki (1975, Monatsh. Math. 80, 277–281) and later called B-products by I. Broere and J. H. Hattingh (1990, Quaest. Math. 13, 191–216) who establish that the lexicographic product of two circulant graphs is again circulant. In this paper, we establish that any B-...

The lexicographic product, a powerful binary operation in graph theory, offers methods for creating novel by establishing connections between each vertex of one and every another. Beyond its fundamental nature, this is found various applications across computer science disciplines, including network analysis, data mining, optimization. In paper, we give definition the weight function to product...

Minimum cycle bases of product graphs can in most situations be constructed from minimum cycle bases of the factors together with a suitable collection of triangles and/or quadrangles determined by the product operation. Here we give an explicit construction for the lexicographic product G ◦ H that generalizes results by Berger and Jaradat to the case that H is not connected.

A tree T , in an edge-colored graph G, is called a rainbow tree if no two edges of T are assigned the same color. A k-rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S ⊆ V (T ). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). G...