The atom-bond connectivity index of chemical bicyclic graphs

The atom-bond connectivity (ABC) index provides a good model for the stabilityof linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC(G) =Σuv∈(G)√ du +dv - 2...     

Atom-bond connectivity index of graphs

The recently introduced atom-bond connectivity (ABC) index has been applied up until now to study the stability of alkanes and the strain energy of cycloalkanes. Furtula et al. (2009) [3] obtained extremal ABC values for chemical trees, and also, it has been shown that the star K_1,n−1, has the maximal ABC value of trees. In this paper, we present the lower and upper bounds on ABC index of graphs and trees, and characterize graphs for which these bounds are best possible.     

The Estrada index of trees

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees with given matching number, independence number, and domination number, respectively. Finally, we give a proof of a conjecture in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) 799-810] on trees with minimum Estrada index among the set of trees with two adjacent vertices of maximum degree.     

Conjugated trees with minimum general Randi? index

The general Randi? index R_α(G) is the sum of the weights (d_G(u)d_G(v))^α over all edges uv of a (molecular) graph G, where α is a real number and d_G(u) is the degree of the vertex u of G. In this paper, for any real number α≤−1, the minimum general Randi? index R_α(T) among all the conjugated trees (trees with a Kekulé structure) is determined and the corresponding extremal conjugated trees are characterized. These trees are also extremal over all the conjugated chemical trees.     

Trees with second minimum general Randić index for α>0

The general Randi? index R _α (G) of a graph G is the sum of the weights (d(u)d(v)) ^α of all edges uv of G, where α is a real number(α≠0) and d(u) denotes the degree of the vertex u. We have known that P _n has minimum general Randi? index for α>0 among trees when n≥5. In this paper, we prove that P _n,3 has second minimum general Randi? index for α>0 among trees when n≥7.     

On the Estrada and Laplacian Estrada indices of graphs

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as , where μ₁,μ₂,…,μ_n are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex. We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.     

Computing topological indices of Sudoku graphs

In QSAR/QSPR study, physico-chemical properties and topological indices such as Randi?, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc. of chemical compounds. Graph theory has found a considerable use in this area of research. The topological indices of certain interconnection networks were studied recently by Imran et al. (Appl Math Comput 244:936–951, 2014). In this paper, we extend this study to \(n\times n\) Sudoku graphs and derive analytical closed results of general Randi? index \(R_{\alpha }(G)\) for different values of “\(\alpha \)” for Sudoku (SK). We also compute the general Randi?, first Zagreb, ABC, GA, \(ABC_{4}\) and \(GA_{5}\) indices and give closed formulae of these indices for Sudoku graphs.     

On Estrada index of trees

Let G be a graph on n vertices, and let λ₁,λ₂,…,λ_n be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ili? and Stevanovi? in Ref. [14].     

On topological properties of poly honeycomb networks

Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as the Randi?, the atom-bond connectivity (ABC) and the geometric-arithmetic (GA) indices are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study poly honeycomb networks which are generated by a honeycomb network of dimension n and derive analytical closed results for the general Randi? index \(R_\alpha (G)\) for different values of \(\alpha \), for a David derived network \((\textit{DD}(n))\) of dimension n, a dominating David derived network \((\textit{DDD}(n))\) of dimension n as well as a regular triangulene silicate network of dimension n. We also compute the general first Zagreb, ABC, GA, \(\textit{ABC}_4\) and \(\textit{GA}_5\) indices for these poly honeycomb networks for the first time and give closed formulas of these degree based indices in case of poly honeycomb networks.     

The smallest Randić index for trees

The general Randi? index R _α (G) is the sum of the weight d(u)d(v) ^α over all edges uv of a graph G, where α is a real number and d(u) is the degree of the vertex u of G. In this paper, for any real number α?≠?0, the first three minimum general Randi? indices among trees are determined, and the corresponding extremal trees are characterized.     

The starlike trees with maximum and minimum second order connectivity index

Let G be a simple graph and h≥0 be an integer. The higher order connectivity index R _h (G) of G is defined as $$R_h(G)=\sum_{v_{i_1}v_{i_2}\cdots v_{i_{h+1}}} \frac{1}{\sqrt {d_{i_1}d_{i_2}\cdots d_{i_{h+1}}}},$$ where d _i denotes the degree of the vertex v _i and $v_{i_{1}}v_{i_{2}}\cdots v_{i_{h+1}}$ runs over all paths of length h in G. A starlike tree is a tree with unique vertex of degree greater than two. Rada and Araujo proved that the starlike trees which have equal connectivity index of order h for all h≥0 are isomorphic. By T(n) we denote the set of the starlike trees on n vertices. In this paper, we characterize the extremal starlike trees with maximum and minimum second order connectivity index in T(n).     

A class of one-step one-stage methods for stiff systems of ordinary differential equations

A new class of one-step one-stage methods (ABC-schemes) designed for the numerical solution of stiff initial value problems for ordinary differential equations is proposed and studied. The Jacobian matrix of the underlying differential equation is used in ABC-schemes. They do not require iteration: a system of linear algebraic equations is once solved at each integration step. ABC-schemes are A- and L-stable methods of the second order, but there are ABC-schemes that have the fourth order for linear differential equations. Some aspects of the implementation of ABC-schemes are discussed. Numerical results are presented, and the schemes are compared with other numerical methods.     

On the eccentric distance sum of graphs

The eccentric distance sum is a novel topological index that offers a vast potential for structure activity/property relationships. For a graph G, it is defined as ξd(G)=_∑v∈Vε(v)D(v), where ε(v) is the eccentricity of the vertex v and D(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. Motivated by [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 934-944], in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum. Various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number and clique number are established. In addition, we present explicit formulae for the values of eccentric distance sum for the Cartesian product, applied to some graphs of chemical interest (like nanotubes and nanotori).     

In search of more triangle centres. A source of classroom projects in Euclidean geometry

A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry.     

A degree bound on decomposable trees

A n-vertex graph is said to be decomposable if for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.     

Broadcasting secure messages via optimal independent spanning trees in folded hypercubes

Fault-tolerant broadcasting and secure message distribution are important issues for network applications. It is a common idea to design multiple spanning trees with a specific property in the underlying graph of a network to serve as a broadcasting scheme or a distribution protocol for receiving high levels of fault-tolerance and security. An n-dimensional folded hypercube, denoted by FQ_n, is a strengthening variation of hypercube by adding additional links between nodes that have the furthest Hamming distance. In, [12], Ho(1990) proposed an algorithm for constructing n+1 edge-disjoint spanning trees each with a height twice the diameter of FQ_n. Yang et al. (2009), [29] recently proved that Ho’s spanning trees are indeed independent, i.e., any two spanning trees have the same root, say r, and for any other node v≠r, the two different paths from v to r, one path in each tree, are internally node-disjoint. In this paper, we provide another construction scheme to produce n+1 independent spanning trees of FQ_n, where the height of each tree is equal to the diameter of FQ_n plus one. As a result, the heights of independent spanning trees constructed in this paper are shown to be optimal.     

The sequence of generalized median triangles and a new shape function

Starting with a triangle ABC and a real number s, we let AA _s , BB _s , CC _s be the cevians that divide the sides BC, CA, AB, respectively, in the ratio s : 1 ? s, and we let ${\mathcal{H}_s(ABC)}$ be the triangle whose side lengths are equal to those of AA _s , BB _s , CC _s . We investigate the sequence of (the shapes of) triangles ${\mathcal{H}_s^n(ABC)}$ , n = 1, 2, ... by introducing a new shape function that suits this sequence. We also use this shape function to prove a theorem of C. F. Parry concerning automedian triangles.     

Algorithmic and structural aspects of the P 3-Radon number

The generalization of classical results about convex sets in ? ⁿ to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the P ₃-convexity on graphs. P ₃-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon’s classical convexity result. We establish hardness results and describe efficient algorithms for trees.