Edge Corona Research Articles

The harmonic index of product graphs

The harmonic index of a graph G is defined as the sum of the weights frac{2}{hbox{deg} _G(u)+hbox{deg} _G(v)} of all edges uv of G, where hbox{deg} _G(u) denotes the degree of a vertex u in G. In this paper, we investigate the harmonic index of Cartesian, lexicographic, tensor, strong, corona and edge corona product of two connected graphs.

Normalized Laplacian spectrum of different type of coronas of two regular graphs

In this paper, we determine the full normalized Laplacian spectrum of the corona, edge corona and neighborhood corona of a connected regular graph with an arbitrary regular graph in terms of the normalized Laplacian eigenvalues of the original graphs. Moreover, applying these results we find some non-regular normalized Laplacian co-spectral graphs.

The normalized Laplacian spectra of the corona and edge corona of two graphs

Let and be two simple graphs. Denote the corona and the edge corona of and by and , respectively. In this paper, first, the normalized Laplacian spectrum of is given in terms of that of and when both and are regular. Then the normalized Laplacian spectrum of is given in terms of that of and for any connected graph and a regular graph . Finally, as applications, the number of spanning trees and the degree Kirchhoff index of and are considered.

On spectra of variants of the corona of two graphs and some new equienergetic graphs

Let G and H be two graphs. The join G ∨ H is the graph obtained by joining every vertex of G with every vertex of H. The corona G ◦ H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H. The neighborhood corona G⋆H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. The edge corona G ⋄H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each terminal vertex of i-th edge of G to every vertex in the i-th copy of H. Let G1, G2, G3 and G4 be regular graphs with disjoint vertex sets. In this paper we compute the spectrum of (G1 ∨ G2) ∪ (G1 ⋆ G3), (G1 ∨ G2) ∪ (G2 ⋆ G3) ∪ (G1 ⋆ G4), (G1∨G2)∪(G1 ◦G3), (G1∨G2)∪(G2 ◦G3)∪(G1 ◦G4), (G1∨G2)∪(G1 ⋄G3), (G1 ∨ G2) ∪ (G2 ⋄ G3) ∪ (G1 ⋄ G4), (G1 ∨ G2) ∪ (G2 ◦ G3) ∪ (G1 ⋆ G3), (G1 ∨ G2) ∪ (G2 ◦ G3) ∪ (G1 ⋄ G4) and (G1 ∨ G2) ∪ (G2 ⋆ G3) ∪ (G1 ⋄ G4). As an application, we show that there exist some new pairs of equienergetic graphs on n vertices for all n ≥ 11.

On the Sparing Number of the Edge-Corona of Graphs

Let $\mathbb{N}_0$ be the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ be its the power set. An integer additive set-indexer (IASI) of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective, where $f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. An integer additive set-indexer $f$ is said to be a weak integer additive set-indexer (weak IASI) if $|f^+(uv)|=\max(|f(u)|,|f(v)|)~\forall ~ uv\in E(G)$. The minimum number of singleton set-labeled edges required for the graph $G$ to admit an IASI is called the sparing number of the graph. In this paper, we discuss the admissibility of weak IASI by a particular type of graph product called the edge corona of two given graphs and determine the sparing number of the edge corona of certain graphs.

The Laplacian spectrum of corona of two graphs

Let G1,G2 be two connected graphs. Denote the corona and the edge corona of G1, G2 by G1 o G2 and G = G1 ◊ G2, respectively. In this paper, we compute the Laplacian spectrum of the corona G o H of two arbitrary graphs G and H and the edge corona of a connected regular graph G1 and an arbitrary graph G2.

The spectrum and the signless Laplacian spectrum of coronae

Let G1,G2 be two simple connected graphs. Denote the corona and the edge corona of G1,G2 by G1∘G2 and G1♢G2, respectively. In this paper, we first introduce a new invariant, the M-coronal of a graph matrix M, where the matrix M is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of G1∘G2 and G1♢G2 in terms of the signless Laplacian spectrum of G1 and G2. In addition, the spectrum of G1♢G2 are also given in terms of the spectrum of G1 and G2. Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs.

The signless Laplacian spectra of the corona and edge corona of two graphs

We give complete information about the signless Laplacian spectrum of the corona of a graph G 1 and a regular graph G 2, and complete information about the signless Laplacian spectrum of the edge corona of a connected regular graph G 1 and a regular graph G 2.

More on Zagreb coindices of graphs

For a nontrivial graph G, its first and second Zagreb coindices are defined, respectively, as M1(G)= ?uv?E(G)(dG (u)+ dG (v)) and M2(G) = ?uv?E dG (u)dG(v), where dG (x) is the degree of vertex x in G. In this paper, we explore further properties of Zagreb coindices. First, we investigate Zagreb coindices of two classes of composite graphs, namely, Mycielski graph and edge corona, and we present explicit formulas for Zagreb coindices of these two composite graphs. Then we we give two estimations on Zagreb coindices of graphs in terms of the number of pendent vertices and Merrifield-Simmons index, respectively. Finally, we give several Nordhaus-Gaddum type bounds for the first Zagreb coindex.

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.

Construction and Behavior of a High Voltage Cylindrical Condenser

Cylinders of lumite glass 35 inches long by 10¼-inch diameter (such as are used on gasoline filling-station pumps) were used for dielectrics, and 0.005-inch aluminum foil for plates. A 7-inch glass margin was left at the ends of the cylinders. Each condenser had a capacity of 0.012 microfarad and would withstand steady voltages of over 100 kilovolts without ``flashing over.'' Sets of rings of 516−inch copper tubing were made to fit the cylinders at the edges of the plates, both inside and out. The rings provided a ``curled'' or ``rounded'' edge, in place of the rather sharp edge of the foil itself. The effect upon the ``edge corona'' and ``flash-over voltage,'' due to various-width bands of aluminum paint on the glass surface adjacent to the metal-plate edges, was tried, with and without the above-mentioned rings, in disruptive discharge and steady-voltage tests. The results showed that the highest voltages could be obtained, without break-over occurring, when the glass margins were chemically clean. It was also found that the best type of plate-edge construction was either the plain aluminum-foil edges without paint, or the curled edges (i.e., with copper rings) with a ¼-inch band of paint.