On inclusive distance vertex irregular labelings (original) (raw)

Local Inclusive Distance Vertex Irregular Graphs

Mathematics, 2021

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

On Distance Irregular Labeling of Disconnected Graphs

Kragujevac Journal of Mathematics

A distance irregular k-labeling of a graph G is a function f : V (G) → {1, 2, . . . , k} such that the weights of all vertices are distinct. The weight of a vertex v, denoted by wt(v), is the sum of labels of all vertices adjacent to v (distance 1 from v), that is, wt(v) = P u∈N(v) f(u). If the graph G admits a distance irregular labeling then G is called a distance irregular graph. The distance irregularity strength of G is the minimum k for which G has a distance irregular k-labeling and is denoted by dis(G). In this paper, we derive a new lower bound of distance irregularity strength for graphs with t pendant vertices. We also determine the distance irregularity strength of some families of disconnected graphs namely disjoint union of paths, suns, helms and friendships.

On inclusive distance vertex irregularity strength of small identical copies of star graphs

Journal of Physics: Conference Series, 2021

For a simple graph G, an inclusive distance vertex irregular k-labeling of G is a mapping λ : V (G) → {1, 2,. .. , k} such that all the vertex-weights are pairwise distinct, where the weight of a vertex v, denoted by wt(v), is the sum of labels of vertices in the close neighborhood of the vertex v. The minimum k for which the graph G has an inclusive distance vertex irregular k-labeling is called the inclusive distance vertex irregularity strength of G, dis(G). Here we introduce a new lower bound for dis(G) and determine the exact value of the inclusive distance vertex irregularity strength for identical copies of star graphs, especially 2Sn and 3Sn.

DISTANCE IRREGULARITY STRENGTH OF GRAPHS WITH PENDANT VERTICES

Opuscula Mathematica, 2022

A vertex k-labeling ϕ : V (G) → {1, 2,. .. , k} on a simple graph G is said to be a distance irregular vertex k-labeling of G if the weights of all vertices of G are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in G. The least integer k for which G has a distance irregular vertex k-labeling is called the distance irregularity strength of G and denoted by dis(G). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.

Total Vertex Irregularity Strength of Some Graphs

2018

Abstract. A vertex irregular total k-labeling of a graph G with vertex setV and edge set E is an assignment of positive integer labels {1,2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted bytvs(G) is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the total vertex irregularity strength of cycle quadrilateral snake, s unflower, double wheel, fungus, triangular book and quadrilateral book.

A Note on Edge Irregularity Strength of Some Graphs

Indonesian Journal of Combinatorics

Let G(V, E) be a finite simple graph and k be some positive integer. A vertex k-labeling of graph G(V,E), Φ : V → {1,2,..., k}, is called edge irregular k-labeling if the edge weights of any two different edges in G are distinct, where the edge weight of e = xy ∈ E(G), wΦ(e), is defined as wΦ(e) = Φ(x) + Φ(y). The edge irregularity strength for graph G is the minimum value of k such that Φ is irregular edge k-labeling for G. In this note we derive the edge irregularity strength of chain graphs mK3−path for m ≢ 3 (mod4) and C[Cn(m)] for all positive integers n ≡ 0 (mod 4) 3n and m. We also propose bounds for the edge irregularity strength of join graph Pm + Ǩn for all integers m, n ≥ 3.

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 Edge Irregular Total k-labeling and Total Edge Irregularity Strength of Barbell Graphs

Let G be a connected graph with a non empty vertex set V (G) and edge set E(G). An edge irregular total k-labeling of a graph G is a labeling λ : V (G) ∪ E(G) → {1, 2, ..., k}, so that every two different edges have different weights. The weight of edge uv of G is the sum of the labels vertices u and v and label of the edge uv, which is can be written as wt(uv) = λ(u) + λ(uv) + λ(v). The total edge irregularity strength of G, denoted by tes(G) is the minimum positive integer k for which the graph G has an edge irregular total k-labeling. Barbell graph B n is obtained by connecting two copies of a complete graph Kn by a bridge. In this research, we determined the total edge irregularity strength of barbell graph Bn for n ≥ 3.