Edge Corona Research Articles

Connected Co-Independent Hop Domination in the Edge Corona and Complementary Prism of Graphs

Let $G$ be a connected graph. A subset $S$ of $V(G)$ is a connected co-independent hop dominating set in $G$ if the subgraph induced by $S$ is connected and $V(G) \backslash S$ is an independent set where for each $v \in V(G) \backslash S$, there exists a vertex $u \in S$ such that $d_G(u,v)=2$. The smallest cardinality of such an $S$ is called the connected co-independent hop domination number of $G$. This paper presents the characterizations of the connected co-independent hop dominating sets in the edge corona and complementary prism of graphs and determines the exact values of their corresponding connected co-independent hop domination number.

Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, \cdots, d_n \rangle$, where $d_1 \ge d_2 \ge \cdots \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{\Delta(G)}a_kx^{k}$, where $a_k$ is the number of vertices of $G$ having degree $k$ for each $i = 1,2,\cdots n = \Delta(G)$. In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.

Closed Geodetic Hop Domination in Graphs

Let G be a simple, undirected and connected graph. A subset S ⊆ V (G) is a geodetic cover of G if IG[S] = V (G), where IG[S] is the set of all vertices of G lying on any geodesic between two vertices in S. A geodetic cover S of G is a closed geodetic cover if the vertices in S are sequentially selected as follows: Select a vertex v1 and let S1 = {v1}. If G is nontrivial, select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Where possible, for i ≥ 3, successively select vertex vi ∈/ IG[Si−1] and let Si = {v1, v2, ..., vi}. Then there exists a positive integer k such that Sk = S. A geodetic cover S of G is a geodetic hop dominating set if every vertex in V (G) \ S is of distance2 from a vertex in S. A geodetic hop dominating set S is a closed geodetic hop dominating set if S is a closed geodetic cover of G. The minimum cardinality of a (closed) geodetic hop dominating set of G is the (closed) geodetic hop domination number of G. This study initiates the study of the closed geodetic hop domination. First, it characterizes all graphs G of order n whose closed geodetic hop domination numbers are 2 or n, and determines the closed geodetic hop domination number of paths, cycles and multigraphs. Next, it shows that any positive integers a and b with 2 ≤ a ≤ b are realizable as the closed geodetic number and closed geodetic hop domination number of a connected graph. Also, every positive integer n, m and k with 4 ≤ m ≤ k and 2k−m+2 ≤ n are realizable as the order, geodetic hop domination number and closed geodetic hop domination number, respectively of a connected graph. Furthermore, the study characterizes the closed geodetic hop dominating sets of graphs resulting from the join, corona and edge corona of graphs.

On (k, ℓ)-locating colorings of graphs

Let c: V(G) → {1, . , ℓ} = [ℓ] be a proper vertex coloring of G and C(i) = {u ∈ V(G): c(u) = i} for i ∈ [ℓ]. The k-color code rk (v|c) of vertex v is the ordered ℓ-tuple (aG (v,C(1)), . , aG (v,C(ℓ))) where If every two vertices have different color codes, then c is a (k, ℓ)-locating coloring of G. The k-locating chromatic number of graph G, denoted by , is the smallest integer ℓ such that G has a (k, ℓ)-locating coloring. In this paper, we propose this concept as an extension of diam(G)-locating chromatic number and 2-locating chromatic number which are known as the locating chromatic number, denoted χL (G), and neighbor-locating chromatic number, denoted , respectively. In this paper, we give sharp bounds for and where G◦H and are the corona and edge corona of G and H, respectively. We formulate an integer linear programming model to determine , noting that almost all graphs have diameter 2 and for every graph G of diameter 2.

Quasi-Laplacian energy of fractal graphs

Graph energy is a measurement of determining the structural information content of graphs. The first Zagreb index can be handled with its connection to graph energy. In this paper, a novel and significant application of the first Zagreb index to composite graphs based on fractal graphs is revealed, and by the relation between quasi-Laplacian energy and the vertex degrees of a graph, we derive closed-form formulas for the quasi-Laplacian energy of fractal graphs or namely R-graphs, R-vertex and edge join, R-vertex and edge corona, R-vertex and edge neighborhood graphs in terms of the corresponding energy, the first Zagreb indices, number of vertices and edges of the underlying graphs.

Convex Roman Dominating Functions on Graphs under some Binary Operations

Let $G$ be a connected graph. A function $f:V(G)\rightarrow \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1 \cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by $\omega_{G}^{CvR}(f)$, is given by $\omega_{G}^{CvR}(f)=\sum_{v \in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by $\gamma_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographicproduct, and Cartesian product of graphs.

Computing the l,k-Clique Metric Dimension of Graphs via (Edge) Corona Products and Integer Linear Programming Model

Let G be a graph with n vertices and CG=X:X is an l-clique of G. A vertex v∈VG is said to resolve a pair of cliques X,Y in G if dGv,X≠dGv,Y where dG is the distance function of G. For a pair of cliques X,Y, the resolving neighbourhood of X and Y, denoted by RGX,Y, is the collection of all vertices which resolve the pair X,Y. A subset S of VG is called an l,k-clique metric generator for G if RGX,Y∩S≥k for each pair of distinct l-cliques X and Y of G. The l,k-clique metric dimension of G, denoted by l−cdimkG, is defined as minS:S is an l,k-clique metric generator of G. In this paper, the l,k-clique metric dimension of corona and edge corona of two graphs are computed. In addition, an integer linear programming model is presented for the l,k-clique metric basis for a given graph G and its l-cliques.

On the antimagicness of generalized edge corona graphs

Given a graph G, a function of assigning distinct labels {1,2,...,|E(G)|} to E(G) such that w(a)≠w(b), ∀ a,b∈V(G) is an antimagic labeling of G where w(a) indicates the vertex sum obtained by summing up all the labels assigned to the edges incident on the vertex a. Let G, Hi, 1≤i≤m be connected graphs such that E(G)={e1,e2,...,em}. A new graph is constructed from G, Hi, 1≤i≤m by adding all possible edges between the end vertices of ei and V(Hi), i∈{1,2,...,m}. The resulting graph is called the generalized edge corona of G and (H1,H2,...,Hm) which is denoted as G⋄(H1,H2,...,Hm). We prove G ⋄ (H1,H2,...,Hm) is antimagic under certain conditions using an algorithmic approach where G has only one vertex of maximum degree three (excluding spider graphs containing uneven legs) and |V(Hi)|≥2, i∈{1,2,...,m}.

Mixed metric dimension over (edge) corona products

A subset S of V(G) is called a mixed resolving set for G if, for every two distinct elements x and y of V ( G )∪ E ( G ) , there exists v ∈ S such that d ( v , x )≠ d ( v , y ) . The mixed metric dimension of G, denoted by dim m ( G ) , is the minimum cardinality of a mixed resolving set in G. In this paper, a closed formula for the mixed metric dimension of corona product of graphs is proved. Also, a sharp upper bound and a closed formula for the mixed metric dimension of edge corona product of graphs is presented.

Global Stable Location-Domination in Graphs

In this paper, we introduce and investigate the concept of global stable location-domination in graphs. We also characterize the global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs and determine the exact value or sharp bound of the corresponding global stable location-domination number.

2-Locating Sets in a Graph

Let $G$ be an undirected graph with vertex-set $V(G)$ and edge-set $E(G)$, respectively. A set $S\subseteq V(G)$ is a $2$-locating set of $G$ if $\big|[\big(N_G(x)\backslash N_G(y)\big)\cap S] \cup [\big(N_G(y)\backslash N_G(x)\big)\cap S]\big|\geq 2$, for all \linebreak $x,y\in V(G)\backslash S$ with $x\neq y$, and for all $v\in S$ and $w\in V(G)\backslash S$, $\big(N_G(v)\backslash N_G(w)\big)\cap S \neq \varnothing$ or $\big(N_G(w)\backslash N_G[v]\big) \cap S\neq \varnothing$. In this paper, we investigate the concept and study 2-locating sets in graphs resulting from some binary operations. Specifically, we characterize the 2-locating sets in the join, corona, edge corona and lexicographic product of graphs, and determine bounds or exact values of the 2-locating number of each of these graphs.

Closeness Centrality of Vertices in Graphs Under Some Operations

In this paper, we revisit the concept of (normalized) closeness centrality of a vertex in a graph and investigate it in some graphs under some operations. Specifically, we derive formulas that compute the closeness centrality of vertices in the shadow graph, complementary prism, edge corona, and disjunction of graphs.

Seidel spectra of some variants of corona operations

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. Recently, there has been a renewed interest in studying Seidel spectrum of graphs and some achievements have been made in this regard. In this paper, we determine the Seidel spectra of some variants of corona operations, such as edge corona, subdivision-vertex corona, subdivision-vertex neighbourhood corona of two graphs and so on. The corresponding Seidel eigenvectors are also described completely. Applying the obtained results, we give some sufficient and necessary conditions for edge corona, subdivision-vertex corona and subdivision-vertex neighbourhood corona of two graphs to be Seidel integral.

Generation of anti-magic graphs from binary graph products

An anti-magic labeling of a graph G is a one-to-one correspondence between E(G) and {1, 2, ··· , |E|} such that the vertex-sum for distinct vertices are different. Vertex-sum of a vertex u ∈ V (G) is the sum of labels assigned to edges incident to the vertex u. It was conjectured by Hartsfield and Ringel that every tree other than K2 has an anti-magic labeling. In this paper, we consider various binary graph products such as corona, edge corona and rooted products to generate anti-magic graphs. We prove that corona products of an anti-magic regular graph G with K1 and K2 are anti-magic. Further, we prove that rooted product of two anti-magic trees are anti-magic. Also, we prove that rooted product of an anti-magic graph with an anti-magic tree admits anti-magic labeling.

On the Aα-spectra of some corona graphs

Let [Formula: see text] be a simple, connected graph. The matrix [Formula: see text] is defined as [Formula: see text], [Formula: see text], where [Formula: see text] is the adjacency matrix and [Formula: see text] is the degree matrix of [Formula: see text]. Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the neighborhood corona, subdivision vertex corona, subdivision edge corona, central vertex corona and central edge corona of two graphs [Formula: see text] and [Formula: see text], respectively. In this paper, we determine the [Formula: see text]-characteristic polynomial and [Formula: see text]-spectra of non-regular graphs obtained from the above operations. As an application, we construct infinitely many pairs of non-regular [Formula: see text]-cospectral graphs. Also, we estimate the [Formula: see text]-energy of the graphs obtained by operating a regular graph [Formula: see text] and a non-regular graph, [Formula: see text].

Correction to: Hitting Times of Random Walks on Edge Corona Product Graphs

Correction to: Hitting Times of Random Walks on Edge Corona Product Graphs

Outer-Connected 2-Resolving Hop Domination in Graphs

. Let G be a connected graph. A set S ⊆ V (G) is an outer-connected 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-resolving hop domination number of G, denoted by γ^c2Rh(G) is the smallest cardinality of an outer-connected 2-resolving hop dominating set of G. This study aims to combine the concept of outer-connected hop domination with the 2-resolving hop dominating sets of graphs. The main results generated in this study include the characterization of outer-connected 2-resolving hop dominating sets in the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values.

Semi-Total Point Graph of Neighbourhood Edge Corona Graph of G and H

A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. Here we concentrated on topological indices involving the number of vertices, the number of edges and the maximum and minimum vertex degree. The aim of this paper is to compute the lower and upper bounds of the second Zagreb index, third Zagreb index, Hyper Zagreb index, Harmonic index, Redefined first Zagreb index, First reformulated Zagreb index, Forgotten topological index, square F-index, Sum-connectivity index, Randic index, Reciprocal Randic index, Gourava index, Sombar index, Nirmala index, Geometric-Arithmetic index and lower bonds of Atom bond connectivity index, Redefined second Zagreb.

Restrained 2-Resolving Hop Domination in Graphs


 
 
 
 Let G be a connected graph. A set S ⊆ V (G) is a restrained 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving hop domination number of G, denoted by γr2Rh(G) is the smallest cardinality of a restrained 2-resolving hop dominating set of G. This study aims to combine the concept of hop domination with the restrained 2-resolving sets of graphs. The main results generated in this study include the characterization of restrained 2-resolving hop dominating sets in the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values.
 
 
 

Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs.