Edge Uv Research Articles

On the Spectral Radius of the Maximum Degree Matrix of Graphs

Let G be a graph with n vertices, and let dG(u) denote the degree of vertex u in G. The maximum degree matrix MG of G is the square matrix of order n whose (u,v)-entry is equal to maxdG(u),dG(v) if vertices u and v are adjacent in G, and zero otherwise. Let Bp,q,r be the graph obtained from the complete graph Kp by removing an edge uv, and identifying vertices u and v with the end vertices u′ and v′ of the paths Pq and Pr, respectively. Let Gn,d denote the set of simple, connected graphs with n vertices and diameter d. A graph in Gn,d that attains the largest spectral radius of the maximum degree matrix is called a maximizing graph. In this paper, we first characterize the spectrum of the maximum degree matrix for graphs of the form Bn−i+2,i,d−i, where 1≤i≤⌊d2⌋. Furthermore, for d≥2, we prove that the maximizing graph in Gn,d is Bn−d+2,⌊d2⌋,⌈d2⌉. Finally, if d≥4 is an even integer, then the spectral radius of the maximum degree matrix in Bn−d+2,⌊d2⌋,⌈d2⌉ can be computed as the largest eigenvalue of a symmetric tridiagonal matrix of order d2+1.

Cordial Labeling in the Context of Some Graphs

Let G=(V,E) be the graph. A mapping f:V→{0,1} is called Binary vertex labeling and f(v) is called the label of the vertex v of G under f . For an edge e=uv, the induced edge labeling f^*:E→{0,1} is given by f^* (e)=|f(u)-f(v)|. Let f:V→{0,1} and for each edge uv, assign the label |f(x) – f(y)|. Then the binary vertex labeling f of a graph G is said to be cordial labeling if |V_f (0)-V_f (1)|≤1 and |e_f (0)-e_f (1)|≤1. In this paper, some graphs are proved for cordial labeling and known to be cordial graph.

Some results on SD-Prime cordial labeling

Given a bijection ʄ : V(G) → {1,2, …,|V(G)|}, we associate 2 integers S = ʄ(u)+ʄ(v) and D = |ʄ(u)-ʄ(v)| with every edge uv in E(G). The labeling ʄ induces an edge labeling ʄ'' : E(G) → {0,1} such that for any edge uv in E(G), ʄ '(uv)=1 if gcd(S,D)=1, and ʄ ' (uv)=0 otherwise. Let eʄ ' (i) be the number of edges labeled with i ∈ {0,1}. We say ʄ is SD-prime cordial labeling if |eʄ '(0)-e ʄ' (1)| ≤ 1. Moreover G is SD-prime cordial if it admits SD-prime cordial labeling. In this paper, we investigate the SD-prime cordial labeling of some derived graphs.

On star-k-PCGs: exploring class boundaries for small k values

A graph G=(V,E) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function w:V→R+ and k mutually exclusive intervals I1,I2,…Ik, such that there is an edge uv∈E if and only if w(u)+w(v)∈⋃iIi. These graphs are related to two important classes of graphs: pairwise compatibility graphs (PCGs) and multithreshold graphs. It is known that for any graph G there exists a k such that G is a star-k-PCG. Thus, for a given graph G it is interesting to know which is the minimum k such that G is a star-k-PCG. We define this minimum k as the star number of the graph, denoted by γ(G). Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of γ(G) for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two-dimensional grid graphs is 2 and that the star number of 4-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

A Study on Odd Prime Labeling of Octopus Graphs Families

An odd prime labeling of a graph G (V,E), is defined as a bijective function f mapping the vertex set V to the set {1,3,5,…,2|V(G)|1}, such that for every edge uvE. the greatest common divisor gcd(f(u),f(v))=1. A graph that permits such a labeling is referred to as an odd prime graph. In this study, we explore the odd prime labeling properties of various graph structures, including the octopus chain graph, octopus ladder graph, twisted octopus ladder graph, and hexa-octopus chain graph.

Mean Cordial Labeling Patterns In Shadow- Graphs Of Paths

Let f be a function from V (G) to {0,1,2}. Give a numerical value (label) from {0, 1, 2} for every edge uv. f is called a mean cordial labeling if |v_f (i)-v_f (j)|≤1 and |e_f (i)-e_f (j)|≤1, ∀ i,j ∈ {0,1,2} where v_f (x) and e_f (x) represent the number of vertices and edges respectively labeled as x(x = 0,1,2). A graph with mean cordial labeling is called a mean cordial graph. In this paper, we study the mean cordial labeling pattern of shadow-graphs of path graphs D_2 (P_n ), for n≥2

The phylogeny number of a graph in the aspect of its triangles and diamonds

The phylogeny number of a graph in the aspect of its triangles and diamonds

An Application for Pair Sum Modulo Labeling in Cryptography

An Application for Pair Sum Modulo Labeling in Cryptography

BALANCED INDEX SETS OF GRAPHS AND SEMIGRAPHS

Let G be a simple graph with vertex set V (G) and edge set E(G). Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. For a graph G(V, E), a friendly labeling f : V (G) → {0, 1} is a binary mapping such that |vf (1) − vf (0)| ≤ 1, where vf (1) and vf (0) represents number of vertices labeled by 1 and 0 respectively. A partial edge labeling f ∗ of G is a labeling of edges such that, an edge uv ∈ E(G) is, f ∗(uv) = 0 if f (u) = f (v) = 0; f ∗(uv) = 1 if f (u) = f (v) = 1 and if f (u)̸ = f (v) then uv is not labeled by f ∗. A graph G is said to be balanced graph if it admits a vertex labeling f that satisfies the conditions, |vf (1) − vf (0)| ≤ 1 and |ef (1) − ef (0)| ≤ 1, where ef (0), ef (1) are the number of edges labeled with 0 and 1 respectively. The balanced index set of the graph G is defined as, {|ef (1) − ef (0)| : the vertex labeling f is friendly}. A semigraph is a generalization of graph. The concept of semigraph was introduced by E. Sampath Kumar. Frank Harrary has defined an edge as a 2-tuple (a, b) of vertices of a graph satisfying, two edges (a, b) and (a′, b′) are equal if and only if either a = a′ and b = b′ or a = b′ and b = a′. Using this notion, E. Sampath Kumar defined semigraph as a pair (V, X) where V is a non-empty set whose elements are called vertices of G and X is a set of n-tuples called edges of G of distinct vertices, for various n ≥ 2 satisfying the conditions: (i) Any two edges of G can have at most one vertex in common; and (ii) two edges (a1, a2, a3, ..., ap) and (b1, b2, b3, ..., bq ) are said to be equal if and only if the number of vertices in both edges must be equal, i.e p = q, and either ai = bi for 1 ≤ i ≤ p or ai = bp−i+1, 1 ≤ i ≤ p. In this article, balance index set of T (Pn), T (Wn), T (Km,n) and T (Sn) determined, and the balance index set of semigraph is introduced. Additionally, the balanced index set of semigraph Cn,m, Kn,m is determined.

GROUP MEAN CORDIAL LABELING OF SOME QUADRILATERAL SNAKE GRAPHS

Let G be a (p, q) graph and let A be a group. Let f : V (G) −→ A be a map. For each edge uv assign the label [o(f (u))+o(f (v)) / 2]. Here o(f (u)) denotes the order of f (u) as an element of the group A. Let I be the set of all integerslabeled by the edges of G. f is called a group mean cordial labeling if the following conditions hold: (1) For x, y ∈ A, |vf (x) − vf (y)| ≤ 1, where vf (x) is the number of vertices labeled with x. (2) For i, j ∈ I, |ef (i) − ef (j)| ≤ 1, where ef (i) denote the number of edges labeled with i. A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take A as the group of fourth roots of unity and prove that, Quadrilateral Snake, Double Quadrilateral Snake, Alternate Quadrilateral Snake and Alternate Double Quadrilateral Snake are groupmean cordial graphs.

EDGE IRREGULAR REFLEXIVE LABELING OF DUMBBELL GRAPH, CORONA OF OPEN LADDER, AND NULL GRAPH

Graph is a simple, connected, undirected graph with vertex set and edge set . A graph is called to have an edge irregular reflexive -labeling if its vertices can be labeled with even numbers from until and its edges can be labeled with positive integers from to such that the weights for all the edges are different, where . The weight of edge uv in graph with labeling, denoted by , is defined as sum of the edge label and all vertex labels incident to that edge. The reflexive edge strength of a graph , denoted by , is the value of minimum of the largest label. In this paper, edge irregular reflexive -labeling for Dumbbell Graph and corona of open ladder and null graph will be determined. The reflexive edge strength of the Dumbbell Graph with and is for and for The reflexive edge strength of the corona of open ladder and null graph with n ≥ 3 and m ≥ 1 is for and for .

Arithmetic Sequential Graceful Labeling For Arrow Graphs

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph

The diameter of sum basic equilibria games

The diameter of sum basic equilibria games

Argon plasma-enhanced UV light emission from ZnO submicrowires grown by hydrothermal method

Argon plasma-enhanced UV light emission from ZnO submicrowires grown by hydrothermal method

On the Extremal Weighted Mostar Index of Bicyclic Graphs

Let G be a simple connected graph with edge set E(G) and vertex set V(G). The weighted Mostar index of a graph G is defined as w+Mo(G)=∑e=uv∈E(G)(dG(u)+dG(v))|nu(e)−nv(e)|, where nu(e) denotes the number of vertices closer to u than to v for an edge uv in G. In this paper, we obtain the upper bound and lower bound of the weighted Mostar index among all bicyclic graphs and characterize the corresponding extremal graphs.

On edge irregularity strength of cycle-star graphs

For a simple graph G, a vertex labeling ϕ : V (G) → {1, 2, . . . , k} is called k-labeling. The weight of an edge uv in G, written wϕ(uv), is the sum of the labels of end vertices u and v, i.e., wϕ(uv) = ϕ(u) + ϕ(v). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two distinct edges u and v, wϕ(u) ̸= wϕ(v). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, written es(G). In this paper, we study the edge irregular k-labeling for cycle-star graph CSk,n−k and determine the exact value for cycle-star graph for 3 ≤ k ≤ 7 and n − k ≥ 1. Finally, we make a conjecture for the edge irregularity strength of CSk,n−k for k ≥ 8 and n − k ≥ 1.

Edge Irregularity Strength of Binomial Trees

For a simple graph G, a vertex labeling φ: V (G) → {1, 2, · · ·, k} is called k- labeling. The weight of an edge uv in G, denoted by wφ(uv), is the sum of the labels of end vertices u and v. A vertex k-labeling is defined to be an edge irregular k- labeling of the graph G if for every two different edges e and f, wφ(e) wφ(f). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we prove that the edge irregularity strength of corona product of a tree T with K1 is es(T K1) = 2es(T). Further, we prove that the edge irregularity strength of binomial trees Bk is 2k−1, for k ≥ 1.

Extensions of results on phylogeny graphs of degree bounded digraphs

Extensions of results on phylogeny graphs of degree bounded digraphs

Pelabelan Super Graceful pada Graf B dan Graf serta Implementasinya pada Matlab

Suppose G is a graph with vertices p dan edges q. Super graceful labeling is a one-to-one function maping on f:V(G) È E(G)  {1,2, … , p + q} such that f(uv) = ½f(u) – f(v)½ differs for each edge uv Î E(G). A graph G is called super graceful if it can be labeled according to the definition of super graceful labeling. The graph B(m,n,k) is a graph consisteing of a path graph of length k conneting the star graph K(1,m) and K(1,n) at the pendant ends. Meanwhile, the graph Pn(1,2,…,n) is a path graph of length by n combining each vertex on the path graph with edges i to i members (1,2,…,n). In this journal, it will be shown that graph B(m,n,k) and graph Pn(1,2,…,n) are super graceful graphs.

On the total edge irregularity strength of certain classes of cycle related graphs

For a graph G=(V,E), an edge irregular total k-labeling is a labeling of the vertices and edges of G with labels from the set {1, 2, ..., k } such that any two different edges have distinct weights. The sum of the label of edge uv and the labels of vertices u and v determines the weight of the edge uv. The smallest possible k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G. We determine the exact value of the total edge irregularity strength for some cycle related graphs.