The total irregularity of a graph G is defined as irrt .G/ D 1 2 P u;v2V.G/ jdG.u/ dG.v/j, where dG.u/ denotes the degree of a vertex u 2 V.G/. In this paper we give (sharp) upper bounds on the total irregularity of graphs under several graph operations including join, lexicographic product, Cartesian product, strong product, direct product, corona product, disjunction and symmetric difference....

The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [3,6], Robotic Navigation [5,8] and Combinatorial Search and Optimization [7]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study it...

We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group AΓ with connected defining graph. We use this to prove that the divergence of AΓ is linear if Γ is a join and quadratic otherwise. As an application, we give a complete description of the cut points in any asymptotic cone of AΓ. We also show that every non-abelia...

A proper total-coloring of graph G is said to be equitable if the number of elements (vertices and edges) in any two color classes differ by at most one, which the required minimum number of colors is called the equitable total chromatic number. In this paper, we prove some theorems on equitable total coloring and derive the equitable total chromatic numbers of Pm ∨ Sn, Pm ∨ Fn and Pm ∨Wn. Keyw...

In this paper, the edge a-Zagreb indices and its coindices of some graph operations, such as generalized hierarchical product, Cartesian Product, join, composition of two graphs are obtained. Using the results obtained here, we deduce the F -indices and its coindices for the above graph operation. Finally, we have computed the edge a-Zagreb Index, F -index and their coindices of some important ...

On-line multicasting has received a lot of attention lately, because of its great practical importance. However, most research assumes undirected graphs, which fail to model the directivity of real networks. In this paper, we ll this gap by ooering upper and lower bounds for the on-line directed multicast problem in a worst case analysis. We examine two types of on-line problems: the join-only ...

The sigma coindex is defined as the sum of squares differences between degrees all nonadjacent vertex pairs. In this paper, we propose some mathematical properties coindex. Later, present precise results for coindices various graph operations such tensor product, Cartesian lexicographic disjunction, strong union, join, and corona product.

Given two graphs G and H, it is said that percolates in H-bootstrap process if one could join all the nonadjacent pairs of vertices some order such a new copy H created at each step. Balogh, Bollobás Morris 2012 investigated threshold percolation Erdős–Rényi model for complete graph proposed similar problem $$H=K_{s,t}$$ , bipartite graph. In this paper, we provide lower upper bounds on $$K_{2,...