Total Edge Irregularity Strength of Arithmetic Book Graphs

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

Given a graph G(V,E) a labeling @ : V ∪E → {1,2,...,k} is called an edge irregular total k-labeling if for every pair of distinct edges uv and xy, @(u)+@(uv)+@(v) 6 @(x)+@(xy)+@(y). The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength. In this paper we consider series composition of uniform theta graphs and obtain its total edge irregularity strength. We have determined the exact value of the total edge irregularity strength of this graph. We have further given an algorithm to prove the result.

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs.

LetG= (V, E)be a finite, simple and undirected graph with a vertex setVand an edge setE. An edge irregular totalk-labelling is a functionf : V ᴗE → {1,2,…,k}such that for any two different edgesxyandx’y’inE, their weights are distinct. The weight of edgexyis the sum of label of edgexy, labels of vertexxand of vertexy. The minimumkfor which the graphGadmits an edge irregular totalk-labelling is called the total edge irregularity strength ofG, denoted bytes(G). We have determined the total edge irregularity strength of book graphs, double book graphs and triple book graphs. In this paper, we show the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs.

In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

Total edge irregularity strength of complete graphs and complete bipartite graphs

Let G be a simple, connected and undirected graph with vertex set V and edge set E. A total k-labeling is defined as totally irregular total k-labeling if the weights of any two different both vertices and edges are distinct. The weight of vertex x is defined as , while the weight of edge xy is . A minimum k for which G has totally irregular total k-labeling is mentioned as total irregularity strength of G and denoted by ts(G). This paper contains investigation of totally irregular total k-labeling and determination of their total irregularity strengths for caterpillar graphs with each internal vertex between two stars has degree three. The results are and for n > 4:

Let G(V, E) be a simple graph. For a labeling ϕ : V (G) ∪ E(G) → {1, 2, …, k} the weight of a vertex x is defined as wt(x) = ϕ(x) + ∑y∈N(x) ϕ (xy), where N(x) is the set of neighbors of x and y. The labeling ϕ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y we have wt(x) ≠ wt(y). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper, we determine total vertex irregularity strengths of trees with maximum degree four and a subdivision of a double-star.

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

A total labeling Φ : V (G) ∪ E(G) → {1, 2, ... , k} is called a vertex irregular total k-labeling of a graph G if different vertices in G have different weights. The weight of a vertex is defined as the sum of the labels of its incident edges and the label of that vertex. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper we deal with the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs. Theta graph is a closer representation of bipolar electric or magnetic fields so labeling of various theta graphs can help the law of physics in future.

Let G = (V, E) is a graph. Total k-labeling on G is a mapping from V ∪ E to {1,2,3,…,k}. The weight of the vertex v ∈ V is the sum of the label of v and the labels of all the edges incident with v. While the weight of the edge uv ∈ E is the sum of the label of uv and the label of u and v. A total k-labeling of G is called a totally irregular total labeling, if the weight of every two distinct vertices are different and the weight of every two distinct edges are different. The minimum k such that a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G). In this paper, we obtain ts of m-copies of rhombus graph

We introduce the notion of total distance irregular labeling, called the local total distance irregular labeling. All edges and vertices are labeled with positive integers 1 to k such that the weight calculated at the vertices induces a vertex coloring if two adjacent vertices has different weight. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of all vertices adjacent and edges incident to $u$ (distance $1$ from $u$). The minimum cardinality of the largest label over all such irregular assignment is called the local total distance irregularity strength, denoted by $tdis_l(G)$. In this paper, we established the lower bound of the local total distance irregularity strength of graphs $G$ and determined exact values of some classes of graphs namely path, cycle, star, bipartite complete, fan and sun graph.

Total labellings, irregularity strengths and colourings

Let H= (T,S), be a finite simple graph, T(H)= T and S(H)= S, respectively, are the sets of vertices and edges on H. Let σ:T∪S→1,2,· · · ,k, be a total k-labeling on H and wσ(x), be a weight of x∈T while using σ labeling, which is evaluated based on the total number of all vertices labels in the neighborhood x and its incident edges. If every x∈T has a different weight, then σ is a distance vertex irregular total k-labeling (DVITL). Total distance vertex irregularity strength of H (tdis(H) is defined as the least k for which H has a DVITL. Our research investigates the DVITL of the path (Pr) and cycle (Cr) graphs. We establish a lower bound and then calculate the precise value of tdis(Pr) and tdis(Cr).

To date, research has examined the physiological determinants of performance in standardized High-Intensity Functional Training (HIFT) workouts but not without the influence of HIFT familiarity. PURPOSE: To examine the predictive value of aerobic fitness, body composition, and total body strength on performance of two standardized HIFT workouts in HIFT-naïve participants. METHODS: Twenty-two recreationally trained individuals (males = 13, females = 9) underwent assessments of peak oxygen consumption (VO2 peak), ventilatory thresholds, body composition, and one repetition maximum tests for the back squat, deadlift, and overhead press in which the sum equaled the total strength. Participants also performed two standardized HIFT workouts: a scaled version of the 19.1 (as many sets of 19 wall balls followed by 19 calories on rower within 15 minutes) and a modified Fran (sets of 21, 15, and 9 repetitions of thrusters and burpees) to determine scores based on total repetitions completed and time-to-completion, respectively. Simple Pearson’s r correlations were used to determine the relationships between HIFT performance variables (19.1 and modified Fran) and the independent variables. A forward stepwise multiple linear regression analysis was performed and significant variables that survived the regression analysis were used to create a predictive model of HIFT performance. RESULTS: Absolute VO2 peak was a significant predictor of 19.1 performance, explaining 39% of its variance (adjusted R2 = 0.39, p = 0.002). For modified Fran, total strength was a significant predictor and explained 33% of the variance in performance (adjusted R2 = 0.33, p = 0.005). CONCLUSION: These results suggest, without any influence of HIFT familiarity or experience, that performance in these two HIFT workouts could be predicted by distinct laboratory-based measurements of fitness.