Computing the Hosoya Polynomial of M-th Level Wheel and Its Subdivision Graph

The determination of Hosoya polynomial is the latest scheme, and it provides an excellent and superior role in finding the Weiner and hyper-Wiener index. The application of Weiner index ranges from the introduction of the concept of information theoretic analogues of topological indices to the use as major tool in crystal and polymer studies. In this paper, we will compute the Hosoya polynomial for multiwheel graph and uniform subdivision of multiwheel graph. Furthermore, we will derive two well-known topological indices for the abovementioned graphs, first Weiner index, and second hyper-Wiener index.

An Algebraic Approach to Find Some Topological Indices of Derived Graphs of the Benzene Ring

Topological indices play a vital role in understanding the chemical and structural properties of the chemical compounds and nanostructures. By finding the M-polynomial of a graph representing a chemical compound, one can obtain the closed forms of some of the commonly known degree-based topological indices of the compound, such as the Zagreb index, general Randic ́ Index and harmonic index. In this article, we obtain the expression for the M-polynomial of the derived graphs of the Benzene ring embedded in the P-type surface network in 2D, namely the line graph, the subdivision graph, and the line graph of its subdivision. Furthermore, some of the degree-based topological indices are obtained for these graphs using their M-polynomials.

Restrained geodetic domination of edge subdivision graph

For a connected graph [Formula: see text], a set [Formula: see text] subset of [Formula: see text] is said to be a geodetic set if all vertices in [Formula: see text] should lie in some [Formula: see text] geodesic for some [Formula: see text]. The minimum cardinality of the geodetic set is the geodetic number. In this paper, the authors discussed the geodetic number, geodetic domination number, and the restrained geodetic domination of the edge subdivision graph.

THE DISTANCE RELATED SPECTRA OF SOME SUBDIVISION RELATED GRAPHS

Let $G$ be a connected graph with a distance matrix $D$. The distance eigenvalues of $G$ are the eigenvalues of $D$, and the distance energy $E_D(G)$ is the sum of its absolute values. The transmission $Tr(v)$ of a vertex $v$ is the sum of the distances from $v$ to all other vertices in $G$. The transmission matrix $Tr(G)$ of $G$ is a diagonal matrix with diagonal entries equal to the transmissions of vertices. The matrices $D^L(G)= Tr(G)-D(G)$ and $D^Q(G)=Tr(G)+D(G)$ are, respectively, the Distance Laplacian and the Distance Signless Laplacian matrices of $G$. The eigenvalues of $D^L(G)$ ( $D^Q(G)$) constitute the Distance Laplacian spectrum ( Distance Signless Laplacian spectrum ). The subdivision graph $S(G)$ of $G$ is obtained by inserting a new vertex into every edge of $G$. We describe here the Distance Spectrum, Distance Laplacian spectrum and Distance Signless Laplacian spectrum of some types of subdivision related graphs of a regular graph in the terms of its adjacency spectrum. We also derive analytic expressions for the distance energy of $\bar{S}(C_p)$, partial complement of the subdivision of a cycle $C_p$ and that of $\overline {S\left( {C_p }\right)}$, complement of the even cycle $C_{2p}$.

On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph

Let \(G=(V,E)\) be a graph. Then the first and second entire Zagreb indices of \(G\) are defined, respectively, as \(M_{1}^{\varepsilon}(G)=\displaystyle \sum_{x \in V(G) \cup E(G)} (d_{G}(x))^{2}\) and \(M_{2}^{\varepsilon}(G)=\displaystyle \sum_{\{x,y\}\in B(G)} d_{G}(x)d_{G}(y)\), where \(B(G)\) denotes the set of all 2-element subsets \(\{x,y\}\) such that \(\{x,y\} \subseteq V(G) \cup E(G)\) and members of \(\{x,y\}\) are adjacent or incident to each other. In this paper, we obtain the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph of the friendship graph.

COHERENCE ANALYSIS FOR ITERATED LINE GRAPHS OF MULTI-SUBDIVISION GRAPH

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence.

Energies of Hypergraphs

In this paper, energies associated with hypergraphs are studied. More precisely, results are obtained for the incidence and the singless Laplacian energies of uniform hypergraphs. In particular, bounds for the incidence energy are obtained as functions of well known parameters, such as maximum degree, Zagreb index and spectral radius. It is also related the incidence and signless Laplacian energies of a hypergraph with the adjacency energies of its subdivision graph and line multigraph, respectively. In addition, the signless Laplacian energy for the class of the power hypergraphs is computed.