Edge Corona Research Articles

On the generalized adjacency, Laplacian and signless Laplacian spectra of the weighted edge corona networks

Many problems in real world, either natural or man-made, can be usefully represented by graphs or networks. Along with a complex topological structure, the weight is a vital factor in characterizing some properties of real networks. In this paper, we define a class of the weighted edge corona product networks. The generalized adjacency (resp., Laplacian and signless Laplacian) spectra with two different structures are determined. As applications, the number of spanning trees and Kirchhoff index of the weighted edge corona product networks are computed.

Some Topological Indices of Edge Corona of Two Graphs

In this paper, we compute the Wiener index, first Zagreb index, second Zagreb index, degree distance index and Gutman index of edge corona of two graphs. Also in some cases we derive formulas for Weiner index, Zagreb indices, degree distance and Gutman index in terms of vertices and edges .

The local metric dimension of edge corona and corona product of cycle graph and path graph

Let G be a nontrivial connected graph with vertex set V(G). For an ordered set W = {w1, w2, …, wn} of n distinct vertices in G, the representation of a vertex v ∈ V(G) with respect to W is an ordered value of distance between v and every vertex of W . The set W is a local metric set of G if the representations of every pair of adjacent vertices with respect to W are different. The local metric set with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G and denoted by diml(G). The edge corona product of cycle graph and path graph denoted by Cm ʘ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining two end-vertices of ith edge of Cm to every vertex in the ith copy of Pn, where 1 ≤ i ≤ m. The corona product of cycle graph and path graph denoted by Cm ◊ Pn, this graph is obtained from a cycle graph Cm and m copies of path graph Pn, and then joining by an edge each vertex from the ith copy of Pn with ith vertex of Cm. In this paper, we determine the local metric dimension of edge corona and corona product of cycle graph and path graph for positive integer m ≥ 3 and n ≥ 1. We obtain the local metric dimension of edge corona product of cycle and path graphs is diml(Cm ◊ Pn)=2 for n = 1 and for n ≥ 2. The local metric dimension of corona product of cycle and path graphs is diml(Cm ◊ Pn)=1 for n = 1 and even positive integer m ≥ 3, diml(Cm ʘ Pn)=2 for n = 1 and odd positive integer m ≥ 3, and for n ≥ 2.

On the strong metric dimension of crossed prism graph and edge corona of cycle with path graph

Let G be a connected graph with vertex set V(G) and edge set E(G). The interval I[u, v] between u and v is defined as the collection of all vertices that belong to some shortest u − v path. A vertex s ∈ V(G) strongly resolves two vertices u and v if u belongs to a shortest v −s path, denoted by u ∈ I[v, s], or v belongs to a shortest u−s path, denoted by v ∈ I[u, s]. A set S ∈ V(G) is a strong resolving set if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension of G, denoted by sdim(G), is defined as the cardinality of the strong metric basis. In this research, we determine the strong metric dimension of crossed prism graph and edge corona graph Cm ◊ Pn. We obtain the strong metric dimension of crossed prism graph with n ≥ 4 is n for n even. The strong metric dimension of edge corona graph Cm ◊ Pn is for m ≥ 4, n = 2, m even; for m ≥ 4, n = 3, m even; for m ≥ 4, n ≥ 3, m even; sdim(Cm ◊Pn)=mn for m ≥ 3, n = 2, m odd; sdim(Cm ◊Pn) = mn−1 for m ≥ 3, n ≥ 3, m odd.

On the metric dimension of generalized broken fan graph and edge corona product of path with star graph

Let G be a connected nontrivial graph with vertex set V(G) and edge set E(G). The distance d(u, v) between two vertices u, v∈V (G) is shortest path from u to v. For an ordered set W = {w1, w2, …, wk}, the representation r(v | W ) of v with respect to W is ordered pair r(v | W ) = (d(v, w1), d(v, w2), …, d(v, wk)). A set W is resolving set of G if r(u | W ) ≠ r(v | W ) for two distinct vertices u, v in G. A resolving set with minimum cardinality is called metric basis of G. The metric dimension of a graph G, denoted by dim(G) is defined as the cardinality of metric basis. In this paper we determine the metric dimension of generalized broken fan BF (a1, a2, …, an) graph and edge corona graph Pm ◊ Sn. We obtained the metric dimension of generalized broken fan graph is for xi = 0 if ∀aj ≡ 0, 2, 4 (mod 5), j < i, and xi = 2 if ∃aj ≡ 1, 3 (mod 5), j < i. The metric dimension of edge corona graph Pm ◊ Sn is dim(Pm ◊ Sn) = n + 1 for m = 2 and 1 ≤ n ≤ 2, dim(Pm ◊ Sn) = 2 for m ≥ 3 and n = 1, dim(Pm ◊ Sn) = m + 1 for m ≥ 3 and n = 2, and dim(Pm ◊ Sn) = n + (m − 2)(n − 2) for m ≥ 2 and n ≥ 3.

The local metric dimension of generalized broken fan graph and edge corona product of star graph and path graph

Let G be a connected nontrivial graph. The distance between two vertices u and v in G is the length of the shortest u-v path, denoted by d(u, v). For an ordered set W = {w1, w2, …, wn} of n vertices on G, the representation of a vertex v with respect to W is ordered pair r(v|W ) = (d(v, w1), d(v, w2), …, d(v, wn)). W is local metric set of G if r(u|W ) ≠ r(v|W ) for every pair of adjacent vertices u and v in G. The local metric set W with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G, denoted by diml(G). In this paper, we determine the local metric dimension of generalized broken fan graph and edge corona product K1,m ◊ Pn graph. We obtained the local metric dimension of generalized broken fan graph is . The local metric dimension of edge corona graph K1,m ◊ Pn is for m = 1 and 1 ≤ n ≤ 5, for m = 1 and n ≥ 6, for m ≥ 2 and 1 ≤ n ≤ 5, and for m ≥ 2 and n ≥ 6.

On Super (a, d) − H− Antimagic Total Labeling of Edge Corona Product and Corona Product of Gear Graph with Path and Cycle

Let graph G consisting of a set of vertices V (G) and a set of edges E(G) be a simple, finite and connected graph. Graph G admits a H− covering, if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. An (a, d) − H− antimagic total labeling of G is a bijective function , such that for all subgraphs H’ that are isomorphic to H. The H− weights φ(H’) = ∑v∈V(H’) ξ(v) + ∑e∈E(H’) ξ(e) constitute an arithmetic progression a, a+d, a+2d, …, a+(k −1)d where a and d are positive integers and k is the number of subgraphs of G isomorphic to H. Moreover, if the smallest possible labels appear on the vertices, then we called G is super (a, d) − H− antimagic. This research aims to study super (a, d) − C4 ʘ Pn antimagic total labeling on Gm ◊ Pn and super (a, d) − C4 ◊ Cn antimagic total labeling on Gm ◊ Cn.

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 of edge corona product on cycle with path graph and cycle with cycle graph

A simple graph G = (V(G), E(G)) admits a 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. An (a, d)-H-antimagic total labeling of G is a bijective function , such that for all subgraphs H’ isomorphic to H, the H’ weights w(H’) = ∑v∈V(H’) ξ(v) + ∑e∈E(H’) ξ(e) constitute an arithmetic progression a, a + d, a + 2d, …, a + (k – 1)d where a and d are positive integers and k is the number of subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices. This research has found super (a, d)-H-antimagic total labeling of edge corona product of cycle and path denoted by Cm ◊ Pn with H is P2 ◊ Pn and super (a, d)-P2 ◊ Cn-antimagic total labeling of Cm ◊ Cn.

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.

Topologies Induced by Neighborhoods of a Graph Under Some Binary Operations

Let G = (V (G), E(G)) be any undirected graph. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G[A] = V (G)\N_G[A], where N_G[A] = A ∪ { x : xa ∈ E(G) for some a ∈ A } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the corona, edge corona, disjunction, symmetric difference, Tensor product, and the strong product of two graphs by determining the subbasic open sets.

Counting independent sets in two class of edge corona product of graphs

We consider the number of independent sets σ(G). In general, the problem of determining the value of σ(G) is NP-complete. For a graph G, the M-S index is defined as the total number of its independent sets. In this paper, we mainly discusses the M-S index of two classes of edge corona product graphs, and the specific expressions are given.

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 relation of and

Every graph G with vertex V and edge E usually referred to as G = (V, E) in this research is a simple, undirected, and non-trivial graph. A set of vertice D of graph G = (V, E) is called Dominating Set if every vertex from is adjacent to a vertex . In this paper, we analyze the relation of domination number of corona product graphs and edge corona product graphs . The difference between corona product and edge corona product is located on the placement. If on corona product, copy of H graph is located at each vertex of graph G, while at edge corona product, copy of H graph is located on each edge of graph G. The difference in the number of vertices and edges between corona product and edge corona product is and . The result of this research is . There is a relation between all domination numbers that have been generated.

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.

Distribution of global defensive k-alliances over some graph products

If $$G=(V_G, E_G)$$ is a graph, then $$S\subseteq V_G$$ is a global defensive k-alliance in G if (i) each vertex not in S has a neighbor in S and (ii) each vertex of S has at least k more neighbors inside S than outside of it. The global defensive k-alliance number of G is the minimum cardinality among all global defensive k-alliance in G. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds is also discussed.

On P2 ⋄ Pn-supermagic labeling of edge corona product of cycle and path graph

A simple graph G = (V, E) admits a H-covering, where H is subgraph of G, 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 of G isomorphic to H and satisfying where m(f) is a constant magic sum. Additionaly, G admits H-supermagic if . The edge corona of Cn and Pn is defined as the graph obtained by taking one copy of Cn and n copies of Pn, and then joining two end-vertices of the i-th edge of Cn to every vertex in the i-th copy of Pn. This research aim is to find H-supermagic covering on an edge corona product of cycle and path graph where H is . We use k-balanced multiset to solve our reserarch. Here, we find that an edge corona product of cycle and path graph is supermagic for n > 3.

Spectra of the generalized edge corona of graphs

Let [Formula: see text] be a simple graph with [Formula: see text] edges and [Formula: see text], [Formula: see text] be [Formula: see text] simple graphs. The generalized edge corona, denoted by [Formula: see text], is the graph obtained by taking one copy of graphs [Formula: see text], [Formula: see text], [Formula: see text] and then joining two end-vertices of the [Formula: see text]th edge [Formula: see text] of [Formula: see text] to every vertex of [Formula: see text] for [Formula: see text]. In this paper, we determine and study the characteristic polynomial, Laplacian polynomial and signless Laplacian polynomial of [Formula: see text]. As an application, we also count the number of spanning trees of the generalized edge corona.

On exponential domination and graph operations

An exponential dominating set of graph $G = (V,E )$ is a subset $Ssubseteq V(G)$ such that $sum_{uin S}(1/2)^{overline{d}{(u,v)-1}}geq 1$ for every vertex $v$ in $V(G)-S$, where $overline{d}(u,v)$ is the distance between vertices $u in S$ and $v in V(G)-S$ in the graph $G -(S-{u})$. The exponential domination number, $gamma_{e}(G)$, is the smallest cardinality of an exponential dominating set. Graph operations are important methods for constructing new graphs, and they play key roles in the design and analysis of networks. In this study, we consider the exponential domination number of graph operations including edge corona, neighborhood corona and power.