Total Eccentricity Research Articles

Total Eccentricity Indices Of A Graph

We have defined and evaluated total eccentricity indices of polyhex nanotubes TUAC6(p, q) and TUZC6(p, q). In thispaper, we compute First Total Eccentricity Index, Second Total Eccentricity Index, First Multiplicative Total Eccentricity Index, Second Multiplicative Total Eccentricity Index, First Hyper Total Eccentricity Index, Second Hyper Total Eccentricity Index, First Multiplicative Hyper Total Eccentricity Index and Second Multiplicative Hyper Total Eccentricity Index of a graph using Total graph of a graph. We evaluate the value of these indices for some standard graphs.

On the non-commuting graph of dihedral group

For a nonabelian group G, the non-commuting graph Γ of G is defined as the graph with vertex-set G - Z ( G ), where Z ( G ) is the center of G , and two distinct vertices of Γ are adjacent if they do not commute in G . In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of the non-commuting graph on D 2 n . We also find the mean distance of the non-commuting graph on D 2 n .

Eccentric topological properties of a graph associated to a finite dimensional vector space

Abstract A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

Further results on Zagreb eccentricity coindices

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.

Computing Eccentricity-Based Topological Indices of 2-Power Interconnection Networks

In a connected graph G with a vertex v, the eccentricity εv of v is the distance between v and a vertex farthest from v in the graph G. Among eccentricity-based topological indices, the eccentric connectivity index, the total eccentricity index, and the Zagreb index are of vital importance. The eccentric connectivity index of G is defined by ξG = ∑v∈VGdvεv, where dv is the degree of the vertex v and εv is the eccentricity of v in G. The topological structure of an interconnected network can be modeled by using graph explanation as a tool. This fact has been universally accepted and used by computer scientists and engineers. More than that, practically, it has been shown that graph theory is a very powerful tool for designing and analyzing the topological structure of interconnection networks. The topological properties of the interconnection network have been computed by Hayat and Imran (2014), Haynes et al. (2002), and Imran et al. (2015). In this paper, we compute the close results for eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index of a hypertree, sibling tree, and X-tree for k-level by using the edge partition method.

Total eccentricity and average eccentricity indices of some chemical graphs: Carbon graphite and copper oxide

In chemistry, topological descriptors have a prominent role; also connectivity indices are very important in chemical graph theory. Chemical graph theory is helpful to understand about the structural properties of a molecular graph. In this article, we study the chemical graphs of copper oxide and carbon graphite and we compute and give close formulas of eccentricity based topological indices, such as total eccentricity index and average eccentricity index for chemical graphs of carbon graphite and copper oxide.

Some Results on Forgotten Topological Coindex

The forgotten topological coindex (also called Lanzhou index) is defined for a simple connected graph G as the sum of the terms du2+dv2 over all non-adjacent vertex pairs uv of G, where du denotes the degree of the vertex u in G. In this paper, we present some inequalities for the forgotten topological coindex in terms of some graph parameters such as the order, size, number of pendent vertices, minimal and maximal vertex degrees, and minimal non-pendent vertex degree. We also study the relation between this invariant and some well-known graph invariants such as the Zagreb indices and coindices, multiplicative Zagreb indices and coindices, Zagreb eccentricity indices, eccentric connectivity index and coindex, and total eccentricity. Exact formulae for computing the forgotten topological coindex of double graphs and extended double cover of a given graph are also proposed.

Eccentric distance sum and adjacent eccentric distance sum index of complement of subgroup graphs of dihedral group

Let G = (V(G),E(G)) is a connected simple graph. Let ec(v) is the eccentricity of vertex v, D(v) = Σu∈V(G) d(u,v) is the sum of all distances from vertex v and deg(v) is the degree of vertex v in G. The eccentric distance sum index of G is defined as ξd(G) = Σv∈V(G)ec(v)D(v) and the adjacent eccentric distance sum index of G is defined as . For positive integer m and m ≥ 3, let D2m be dihedral group of order 2m and N is a normal subgroup of D2m. The subgroup graph ΓN(D2m) of dihedral group D2m is a simple graph with vertex set D2m and two distinct vertices x and y are adjacent if and only if xy ∈ N. In the present paper, we compute eccentric distance sum and adjacent eccentric distance sum index of complement of subgroup graph of dihedral group D2m. Total eccentricity, eccentric connectivity index, first Zagreb index, and second Zagreb index of these graphs are also determined.

Eccentricity based topological indices of honeycomb networks

Graph theory has made much advancement in the field of mathematical chemistry in last few decades. Now a days, chemical graph theory has become very popular among researchers because of its wide range of applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of honeycomb network denoted by HCn and compute the total eccentricity index, average eccentricity index, eccentricity based Zagreb indices, atom-bond connectivity index and geometricarithmetic index of honeycomb networks. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies of this netwrok.

Some Eccentricity-Based Topological Indices and Polynomials of Poly(EThyleneAmidoAmine) (PETAA) Dendrimers

Topological indices have been computed for various molecular structures over many years. These are numerical invariants associated with molecular structures and are helpful in featuring many properties. Among these molecular descriptors, the eccentricity connectivity index has a dynamic role due to its ability of estimating pharmaceutical properties. In this article, eccentric connectivity, total eccentricity connectivity, augmented eccentric connectivity, first Zagreb eccentricity, modified eccentric connectivity, second Zagreb eccentricity, and the edge version of eccentric connectivity indices, are computed for the molecular graph of a PolyEThyleneAmidoAmine (PETAA) dendrimer. Moreover, the explicit representations of the polynomials associated with some of these indices are also computed.

Eccentric connectivity coindex under graph operations

The eccentric connectivity coindex is defined as the total eccentricity sum of all non-adjacent vertex pairs in a connected graph. In this paper, we present exact expressions or sharp lower bounds for the eccentric connectivity coindex of several graph operations such as sum, disjunction, symmetric difference, lexicographic product, generalized hierarchical product, Cartesian product, rooted product, corona product, and strong product.

Complexity of the Szeged index, edge orbits, and some nanotubical fullerenes

Let I be a summation-type topological index. The I-complexity CI (G) of a graph G is the number of different contributions to I(G) in its summation formula. In this paper the complexity CSz(G) is investigated, where Sz is the well-studied Szeged index. Let Oe(G) (resp. Ov(G)) be the number of edge (resp. vertex) orbits of G. While CSz(G)  Oe(G) holds for any graph G, it is shown that for any m  there exists a vertex-transitive graph Gm with CSz(Gm) = Oe(Gm) = m. Also, for any \le k \le m + 1 there exists a graph Gm,k with CSz(Gm,k) = Oe(Gm,k) = m and CW(Gm,k) = Ov(Gm,k) = k. The Sz-complexity is determined for a family of (5,0)-nanotubical fullerenes and the Szeged index is compared with the total eccentricity.

Computation of total eccentricity using python program

In this manuscript, we use Python program to compute the total eccentricity of various bridge molecular graph …

Eccentricity-Based Topological Indices of a Cyclic Octahedron Structure

In this article, we study the chemical graph of a cyclic octahedron structure of dimension n and compute the eccentric connectivity polynomial, the eccentric connectivity index, the total eccentricity, the average eccentricity, the first Zagreb index, the second Zagreb index, the third Zagreb index, the atom bond connectivity index and the geometric arithmetic index of the cyclic octahedron structure. Furthermore, we give the analytically closed formulas of these indices which are helpful for studying the underlying topologies.

Eccentricity Based Topological Indices of an Oxide Network

Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity (ABC) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies.

Forced oscillations and stability analysis of a nonlinear micro-rotating shaft incorporating a non-classical theory

In this paper, the stability and bifurcation analysis of symmetrical and asymmetrical micro-rotating shafts are investigated when the rotational speed is in the vicinity of the critical speed. With the help of Hamilton’s principle, nonlinear equations of motion are derived based on non-classical theories such as the strain gradient theory. In the dynamic modeling, the geometric nonlinearities due to strains, and strain gradients are considered. The bifurcations and steady state solution are compared between the classical theory and the non-classical theories. It is observed that using a non-classical theory has considerable effect in the steady-state response and bifurcations of the system. As a result, under the classical theory, the symmetrical shaft becomes completely stable in the least damping coefficient, while the asymmetrical shaft becomes completely stable in the highest damping coefficient. Under the modified strain gradient theory, the symmetrical shaft becomes completely stable in the least total eccentricity, and under the classical theory the asymmetrical shaft becomes completely stable in the highest total eccentricity. Also, it is shown that by increasing the ratio of the radius of gyration per length scale parameter, the results of the non-classical theory approach those of the classical theory.

Diagnosis of air‐gap eccentricity and partial demagnetisation of an interior permanent magnet synchronous motor based on inverse transient complex inductance vector theory

On the basis of predecessors, a theory named inverse transient complex inductance vector is proposed for air-gap eccentricity fault diagnosis of interior permanent magnet synchronous motors, and a specific voltage sequence injection method with simple implementation is proposed here. It is a real-time visualised diagnostic approach. A unique fault index is proposed to diagnose eccentricities. The absolute value of the fault index is positively correlated with the total eccentricity degree. Three types of artificial eccentricities and one type of artificial partial demagnetisation are tested. Both simulations and experiments match the proposed theory and diagnostic approach. The approach gives a clear distinction in detecting air-gap eccentricity and partial magnet demagnetisation. It performs best in the low-speed operating condition.

Further results on non-self-centrality measures of graphs

For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E3(G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E3(G) = ?uv?E(G)|?G(u)-?G(v)| and N(G) = ? {u,v}?V(G) |?G(u)-?G(v)|, where ?G(u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the nonself-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-selfcentrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.

Eccentric connectivity polynomial and Total eccentricity polynomial of \(NA^n_m\) Nanotube

Let \(G\) be a molecular graph with vertex set \(V(G)\) and edge set \(E(G)\). In chemical graph theory, for a molecular graph we have many invariant polynomials and topological indices. The length of a shortest path between two vertices of \(G\) is called distance. In a connected graph \(G\), the eccentricity \(\epsilon(v)\) of vertex \(v\) is the distance between \(v\) and a vertex farthest from \(v\) in \(G\). In this paper, we consider \(NA^n_m\) nanotube and compute eccentric connectivity polynomial and total eccentricity polynomial. Furthermore, we also compute some eccentricity based Zagreb indices of \(NA^n_m\).

Study of Dendrimers by Topological Indices

Abstract In this paper, five degree based topological indices, the first Zagreb (M 1), second Zagreb (M 2), first multiple Zagreb (PM 1), second multiple Zagreb (PM 2), and the hyper Zagreb (HM) indices of two types of dendrimers are studied. In addition, two distance based topological indices, the total eccentricity (θ) and eccentric connectivity (ξc ) indices of these dendrimers are computed.