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Title: Entire Irregularity Strengths and Cordial Labelings of graphs
Authors: Javed, Aisha
Keywords: Physical Sciences
Issue Date: 2019
Publisher: Government College University, Lahore.
Abstract: “An edge-covering of G is a family of subgraphs H1, H2, . . . , Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi , i = 1, 2, . . . , t. Then it is said that G admits an (H1, H2, . . . , Ht)-(edge) covering. If every subgraph Hi is isomorphic to a given graph H, then the graph G admits an H-covering.” “Let G = (V, E, F) be a plane graph admitting H-covering. For the subgraph H ⊆ G under an entire k-labeling ϕ : V (G) ∪ E(G) ∪ F(G) → {1, 2, . . . , k}, we define the associated H-weight as wϕ(H) = X v∈V (H) ϕ(v) + X e∈E(H) ϕ(e) + X f∈F(H) ϕ(f). ” “An entire k-labeling ϕ is called an H-irregular entire k-labeling of the plane graph G if for every two different subgraphs H0 and H00 isomorphic to H there is wϕ(H0 ) 6= wϕ(H00). The entire H-irregularity strength of a plane graph G, denoted by Ehs(G, H), is the smallest integer k such that G has an H-irregular entire k-labeling.” Similarly is defined a vertex-face He-irregularityestrength denotedeby vfhs(G, H) and also an edge-face H-eirregularityestrength denotedeby efhs(G, H). One of the interesting kind of labelings are cordial labelings. For a simple graph G = (V, E) we deal with “an edge labeling ϕ : E(G) → {0, 1, . . . , k − 1}, 2 ≤ k ≤ |E(G)|, which induces a vertex labeling ϕ ∗ : V (G) → {0, 1, . . . , k − 1} in such a way that for each vertex v, assigns the label ϕ(e1) · ϕ(e2) · . . . · ϕ(en) (mod k), where e1, e2, . . . , en are the edges incident to the vertex v.” “The labeling ϕ is called a k-total edge product cordial labeling of G if |(eϕ(i) + vϕ∗ (i)) − (eϕ(j) + vϕ∗ (j))| ≤ 1 for every i, j, 0 ≤ i < j ≤ k − 1, where eϕ(i) and vϕ∗ (i) is the number of edges and vertices with ϕ(e) = i and ϕ ∗ (v) = i, respectively.” In the thesis, we estimate the bounds of the parameters Ehs(G, H), vfhs(G, H) and efhs(G, H), and determine the precise values of these parameters for certain families of plane graphs to demonstrate that the obtained bounds are tight. Also we examine the existence of 3-totaleedge productecordialelabelings for rhombic grid graphs, for toroidal fullerenes and for Klein-bottle fullerenes.
Gov't Doc #: 20939
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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