Comparing the irregularity and the total irregularity of graphs (original) (raw)
The total irregularity of graphs under graph operations
Miskolc Mathematical Notes, 2014
The total irregularity of a graph G is defined as irr t (G) = 1 2 u,v∈V (G) |d G (u)−d G (v)|, where d G (u) denotes the degree of a vertex u ∈ 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.
On vertex and edge H-irregularity strengths of graphs
Discrete Mathematics, Algorithms and Applications, 2016
We introduce two new graph characteristics, the edge [Formula: see text]-irregularity strength and the vertex [Formula: see text]-irregularity strength of a graph. We estimate the bounds of these parameters and determine their exact values for several families of graphs namely, paths, ladders and fans.
An iterative approach to graph irregularity strength
Discrete Applied Mathematics, 2010
An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s(T ) for trees T in which every two vertices of degree not equal to two are at distance at least eight.
Graphs with Equal Irregularity Indices
Acta Polytechnica Hungarica, 2014
The irregularity of a graph can be defined by different so-called graph topological indices. In this paper, we consider the irregularities of graphs with respect to the Collatz-Sinogowitz index [8], the variance of the vertex degrees [6], the irregularity of a graph [4], and the total irregularity of a graph [1]. It is known that these irregularity measures are not always compatible. Here, we investigate the problem of determining pairs or classes of graphs for which two or more of the above mentioned irregularity measures are equal. While in [17] this problem was tackled in the case of bidegreed graphs, here we go a step further by considering tridegreed graphs and graphs with arbitrarily large degree sets. In addition we pressent the smallest graphs for which all above irregularity indices are equal.
Bulletin of the InternationalMathematical Virtual Institute, 2019
A totally irregular total k-labeling f : V ∪ E → {1, 2, 3,. .. , k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and y their vertex-weights wt h (x) = wt h (y) where the vertex-weight wt h (x) = h(x) + xy∈E h(xz) and also for every two different edges xy and x ′ y ′ of G their edge-weights wt h (xy) = h(x) + h(xy) + h(y) and wt h (x ′ y ′) = h(x ′) + h(x ′ y ′) + h(y ′) are distinct. A total irregularity strength of graph G, denoted by ts(G) is defined as the minimum k for which a graph G has a totally irregular total k-labeling. In this paper, we investigate double fan, double triangular snake, joint-wheel and Pm + Km whose total irregularity strength equals to the lower bound. 2010 Mathematics Subject Classification. 05C78. Key words and phrases. vertex irregular total k-labeling; edge irregular total k-labeling; total irregularity strength;double fan graph;double triangular snake graph; joint-wheel graph.
The irregularity of graphs under graph operations
Discussiones Mathematicae Graph Theory, 2014
The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = uv∈E(G) |d G (u) − d G (v)|, where d G (u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. We give exact expressions or (sharp) upper bounds on the irregularity of graphs under the above mentioned operations.
Further results on edge irregularity strength of graphs
Indonesian Journal of Combinatorics
A vertex k-labelling φ : V (G) −→ {1, 2,. .. , k} is called irregular k-labeling of the graph G if for every two different edges e and f , there is w φ (e) = w φ (f); where the weight of an edge is given by e = xy ∈ E(G) is w φ (xy) = φ(x) + φ(y). The minimum k for which the graph G has an edge irregular k-labelling is called edge irregularity strength of G, denoted by es(G). In the paper, we determine the exact value of the edge irregularity strength of caterpillars, n-star graphs, (n, t)-kite graphs, cycle chains and friendship graphs.
On the total vertex irregularity strength of trees
Discrete Mathematics, 2010
A vertex-irregular total k-labelling λ : V (G) ∪ E(G) −→ {1, 2, ..., k} of a graph G is a labelling of vertices and edges of G in such a way that for any different vertices x and y, their weights wt(x) and wt(y) are distinct. The weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x. The minimum k for which a graph G has a vertex-irregular total k-labelling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper, we determine the total vertex irregularity strength of trees.
Graphs with maximal irregularity
Filomat, 2014
Albertson [3] has defined the P irregularity of a simple undirected graph G = (V,E) as irr(G) =?uv?E |dG(u)- dG(v)|, where dG(u) denotes the degree of a vertex u ? V. Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [12]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n3/27: Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound ?n/3? ?2n/3? (?2n/3? -1) is sharp. We also present lower bounds on the maximal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.
On edge irregularity strength of graphs
Applied Mathematics and Computation, 2014
An edge irregular total k-labeling of a graph G is a labeling of the vertices and edges with labels 1, 2, . . . , k such that the weights of any two different edges are distinct, where the weight of an edge is the sum of the label of the edge itself and the labels of its two end vertices. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength, tes(G). In this paper we determine the exact values of the total edge irregularity strength of zigzag graphs.