Copies Of G2 Research Articles

On H-supermagic labeling of friendship edge corona product with path and sun edge corona product with path

Let G be a simple, finite, connected and undirected graph with vertex set V and edge set E. Graph G admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. Graph G is H− magic if there is a total labeling , such that each subgraph H′ = (V′, E′) of G isomorphic to H and satisfying where m(f) is a constant magic sum. Additionaly, G admits H− supermagic if f(V ) = {1, 2, …, |V(G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research provides C3 ◊ Pn-supermagic labeling on fn ◊ Pn and P3 ◊ Pn-supermagic labeling on Sn ◊ Pn.

Super (a, d)-H-antimagic total labeling on wheel edge corona product with a path and a cycle

A simple graph G = (V (G), E(G)) admits an H-covering, where H is subgraph of G, if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. A graph G is said to be an (a, d)-H-antimagic covering if there exist a bijective function λ : V(G) → E(G) → {1, 2, …, |V(G)| + |E(G)|}, such that for all subgraphs H’ isomorphic to H, the H-weights w(H’) = ∑v∈V(H’) λ(v) + ∑e∈E(H’) λ(e) constitute an arithmetic progression a, a + d, a + 2d, …, a + (t − 1)d where a and d are positive integers and t is the number of subgraphs of G isomorphic to H. If the vertices contain the smallest possible labels, we called super (a, d)-H-antimagic. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research aims to find (a, d)-H-antimagic total labeling on Wn ◊ Pn with H is C3 ◊ Pn and Wn ◊ Cn with H is C3 ◊ Cn.

H− Supermagic labeling on domino ʘ Pn and Cn ◊ Sn

A finite simple graph G with vertex set V and edge set E admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. We said G to be an H magic if there exists a total labeling , such that for each subgraph H’ = (V’, E’) of G isomorphic to H, ∑v∈V f(v) + ∑e∈E f(e) = m(f) where m(f) is a constant magic sum. Then G is H supermagic if f(V ) = {1, 2, …, |V (G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research focuses on C4 ʘ Pn− supermagic labeling on Domino ʘ Pn and P2 ◊ Sn supermagic labeling on Cn ◊ Sn.

On the power domination number of corona product and join graphs

A set D of vertices of graph G is called a dominating set if every vertex is adjacent to some vertex . A set D to be a power dominating set of a graph if every vertex and every edge in the system are monitored by the set D. The power domination number γP (G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we analyze the power domination number of corona graphs and join graphs. The corona product of two graphs G1 and G2 denoted by is defined as the graph G obtained by taking one copy of G1 and copies of G2, and then joined by an edge the i’th vertex of G1 to every vertex in the i’th copy of G2. The join of two graphs H1 and H2 is a graph formed from disjoint copies of H1 and H2 by connecting every vertex of H1 to every vertex of H2. Join graph H1 and H2 denoted by H1 + H2. The results show that the power domination number of some corona product and join graphs attain the lower bound.

On the partition dimension of edge corona product of path and cycle

Let v be a vertex of a connected graph G(V, E). Let S be a subset of V (G). For an ordered partition of V (G), the representation of a vertex with respect to is the k-vectors , where d(v, Sk) represents the distance between the vertex v and the set Sk, defined by . The partition of V (G) is a resolving partition set if the k-vektors are distinct. The minimum resolving partition is a partition dimension of G, denoted by pd(G). The edge corona of G1 and G2 is defined as the graph obtained by taking one copy of G1 and copies of G2 and then joining two end-vertices of the i-th edge of G1 to every vertex in the i-th copy of G2. In this paper, we will study the partition dimension of the edge corona of cycle and path, namely and for n ≤ 2 and m ≤ 3.

The rainbow vertex connection number of edge corona product graphs

Let G1, G2 be a special graphs with vertices of G1 1,2,…, n and edges of G1 1,2,… m. The generalized edge corona product of graphs G1 and G2, denoted by G1 ⋄ G1 is obtained by taking one copy of graph G1 and m copy of G2, thus for each edge ek = ij of G, joining edge between the two end-vertices i, j of ek and each vertex of the k-copy of G2. A rainbow vertex-coloring graph G where adjacent vertices u-v and its internal vertices have distinct colors. A path is called a rainbow path if no two verticess of the path have the same color. A rainbow vertex-connection number of graph G is minimun number of colors in graph G to connected every two distinct internal vertices u and v such that a graph G naturally rainbow vertex-connected, denoted by rvc (G). In this paper, we determine minimum integer for rainbow vertex coloring of edge corona product on cycle and path such as Pn ⋄ Pm, Pn ⋄ Cm, Cn ⋄ Pm, and Cn ⋄ Cm.

More on the total dominator chromatic number of a graph

Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G1 and G2 is denoted by G1 ⋆ G2 and is the graph obtained by taking one copy of G1 and |V(G1)| copies of G2, and joining the neighbours of the ith vertex of G1 to every vertex in the ith copy of G2. In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the TDC-number of G. We study the stability and bondage number of certatin graphs.

Polarity graphs and Ramsey numbers for [formula omitted] versus stars

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) shows that R(C4,K1,n)≤n+⌊n−1⌋+2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R(C4,K1,q2−t)=q2+q−(t−1) if q is an odd prime power, 1≤t≤2⌈q4⌉ and t≠2⌈q4⌉−1, which extends a result on R(C4,K1,q2−t) obtained by Parsons (1976).

Some values of Ramsey numbers for C4 versus stars

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) [6] shows that R(C4,K1,n)≤n+⌊n−1⌋+2 for all n≥2 and the equality holds if n is the square of a prime power. Let q be a prime power. In this paper, we first construct a graph Γq on q2−1 vertices without C4 by using the Galois field Fq, and then we prove that R(C4,K1,(q−1)2+t)=(q−1)2+q+t for q≥4 is even and t=1,0,−2, and R(C4,K1,q(q−1)−t)=q2−t for q≥5 is odd and t=2,4,...,2⌈q4⌉.

Distance spectrum of Indu–Bala product of graphs

The D-eigenvalues μ1,μ2,…,μn of a graph G of order n are the eigenvalues of its distance matrix D and form the distance spectrum or D-spectrum of G denoted by SpecD(G). Let G1 and G2 be two regular graphs. The Indu–Bala product of G1 and G2 is denoted by G1▾G2 and is obtained from two disjoint copies of the join G1∨G2 of G1 and G2 by joining the corresponding vertices in the two copies of G2. In this paper we obtain the distance spectrum of G1▾G2 in terms of the adjacency spectra of G1 and G2. We use this result to obtain a new class of distance equienergetic graphs of diameter 3. We also prove that the class of graphs Kn¯▾Kn+1¯ has integral distance spectrum.

Further Results on Resistance Distance and Kirchhoff Index in Electric Networks

In electric circuit theory, it is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index. LetQ(G)be the graph obtained fromGby inserting a new vertex into every edge ofGand by joining by edges those pairs of these new vertices which lie on adjacent edges ofG. The set of such new vertices is denoted byI(G). TheQ-vertex corona ofG1andG2, denoted byG1⊙QG2, is the graph obtained from vertex disjointQ(G1)andVG1copies ofG2by joining theith vertex ofV(G1)to every vertex in theith copy ofG2. TheQ-edge corona ofG1andG2, denoted byG1⊖QG2, is the graph obtained from vertex disjointQ(G1)andIG1copies ofG2by joining theith vertex ofI(G1)to every vertex in theith copy ofG2. The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such asQ-vertex corona andQ-edge corona networks.

The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs

Let R(G) be the graph obtained from G by adding a new vertex corresponding to each edge of G and by joining each new vertex to the end vertices of the corresponding edge. Let I(G) be the set of newly added vertices. The R-vertex corona of G1 and G2, denoted by G1⊙G2, is the graph obtained from vertex disjoint R(G1) and |V(G1)| copies of G2 by joining the ith vertex of V(G1) to every vertex in the ith copy of G2. The R-edge corona of G1 and G2, denoted by G1⊖G2, is the graph obtained from vertex disjoint R(G1) and |I(G1)| copies of G2 by joining the ith vertex of I(G1) to every vertex in the ith copy of G2. Liu et al. gave formulae for the Laplacian polynomial and Kirchhoff index of RT(G) in [19]. In this paper, we give the Laplacian polynomials of G1⊙G2 and G1⊖G2 for a regular graph G1 and an arbitrary graph G2; on the other hand, we derive formulae and lower bounds of Kirchhoff index of these graphs and generalize the existing results.

Clusters and various versions of wiener-type invariants

The Wiener type invariant W(λ)(G) of a simple connected graph G is defined as the sum of the terms d(u, ν ׀G)λ over all unordered pairs {u,ν} of vertices of G, where d(u, ν ׀G)λ denotes the distance between the vertices u and v in G and λ is an arbitrary real number. The cluster G1{G2} of a graph G1 and a rooted graph G2 is the graph obtained by taking one copy of G1 and |V (G1)| copies of G2, and by identifying the root vertex of the i-th copy of G2 with the i-th vertex of G1, for i = 1; 2; … ; |V (G1)|. In this paper, we study the behavior of three versions of Wiener type invariant under the cluster product. Results are applied to compute several distance-based topological invariants of bristled and bridge graphs by specializing components in clusters.

Spectra of subdivision-vertex and subdivision-edge neighbourhood coronae

Let G=(V(G),E(G)) be a graph with vertex set V(G) and edge set E(G). The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G1 and G2 be two vertex disjoint graphs. The subdivision-vertex neighbourhood corona of G1 and G2, denoted by G1G2, is the graph obtained from S(G1) and |V(G1)| copies of G2, all vertex disjoint, and joining the neighbours of the ith vertex of V(G1) to every vertex in the ith copy of G2. The subdivision-edge neighbourhood corona of G1 and G2, denoted by G1⊟G2, is the graph obtained from S(G1) and |I(G1)| copies of G2, all vertex disjoint, and joining the neighbours of the ith vertex of I(G1) to every vertex in the ith copy of G2, where I(G1) is the set of inserted vertices of S(G1). In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of G1G2 (respectively, G1⊟G2) in terms of the corresponding spectra of G1 and G2. As applications, these results enable us to construct infinitely many pairs of cospectral graphs, and using the results on the Laplacian spectra of subdivision-vertex neighbourhood coronae, new families of expander graphs are constructed from known ones.

The spectrum of the edge corona of two graphs

Given two graphs G1, with vertices 1, 2, ..., n and edges e1, e2, ..., em, and G2, the edge corona G1 | G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i, j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 | G2 are given in terms of the spectrum and Laplacian spectrum of G1 and G2, respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

Linear Ramsey numbers of sparse graphs

AbstractThe Ramsey number R(G1,G2) of two graphs G1 and G2 is the least integer p so that either a graph G of order p contains a copy of G1 or its complement Gc contains a copy of G2. In 1973, Burr and Erdős offered a total of $25 for settling the conjecture that there is a constant c = c(d) so that R(G,G)≤ c|V(G)| for all d‐degenerate graphs G, i.e., the Ramsey numbers grow linearly for d‐degenerate graphs. We show in this paper that the Ramsey numbers grow linearly for degenerate graphs versus some sparser graphs, arrangeable graphs, and crowns for example. This implies that the Ramsey numbers grow linearly for degenerate graphs versus graphs with bounded maximum degree, planar graphs, or graphs without containing any topological minor of a fixed clique, etc. © 2005 Wiley Periodicals, Inc. J Graph Theory