Zehui Shao

On the maximum ABC index of graphs without pendent vertices

Abstract Let G be a simple graph. The atom–bond connectivity index (ABC) of G is defined as A B C ( G ) = ∑ u v ∈ E ( G ) d ( u ) + d ( v ) − 2 d ( u ) d ( v ) , where d(v) denotes the degree of vertex v of G. We characterize the graphs with n vertices, minimum vertex degree  ≥ 2, and m edges for m = 2 n − 4 and m = 2 n − 3 , that have maximum ABC index.

Computing Zagreb Indices and Zagreb Polynomials for Symmetrical Nanotubes

Topological indices are numbers related to sub-atomic graphs to allow quantitative structure-movement/property/danger connections. These topological indices correspond to some specific physico-concoction properties such as breaking point, security, strain vitality of chemical compounds. The idea of topological indices were set up in compound graph hypothesis in view of vertex degrees. These indices are valuable in the investigation of mitigating exercises of specific Nanotubes and compound systems. In this paper, we discuss Zagreb types of indices and Zagreb polynomials for a few Nanotubes covered by cycles.

A genetic algorithm for solving multi-constrained function optimization problems based on KS function

In this paper, a new genetic algorithm for solving multi-constrained optimization problems based on KS function is proposed. Firstly, utilizing the agglomeration features of KS function, all constraints of optimization problems are agglomerated to only one constraint. Then, we use genetic algorithm to solve the optimization problem after the compression of constraints. Finally, the simulation results on benchmark functions show the efficiency of our algorithm.

Integer linear programming model and satisfiability test reduction for distance constrained labellings of graphs: the case of L(3,2,1)labelling for products of paths and cycles

Let u and v be vertices of a graph G = ( V , E ) and d ( u , v ) be the distance between u and v in G . For positive integers k 1 , k 2 , ... , k n with

Upper bounds on the connection probability for 2-D meshes and tori

k_1 \gt k_2 \gt \cdots \gt k_n

On dominating sets of maximal outerplanar and planar graphs

k 1 >k 2 >⋯>k n an L ( k 1 , k 2 , ... , k n )-labelling of G is a function f : V ( G ) → {0, 1, ... } such that for every u , v ∈ V ( G ) and for all 1 ≤ i ≤ n , |f ( u ) - f ( v ) | ≥ k i if d ( u , v ) = i . The span of f is the difference between the largest and the smallest numbers in f ( V ( G )). The

A note on the chromatic number of the square of the Cartesian product of two cycles

\lambda _{k_1 \comma k_2 \comma \ldots \comma k_n }

More Constructive Lower Bounds on Classical Ramsey Numbers

λk 1 ,k 2 ,...,k n -number of G is the minimum span over all L ( k 1 , k 2 , ... , k n )-labellings of G . In this study, an integer linear programming model and a satisfiability test reduction for an L ( k 1 , k 2 , ... , k n )-labelling are proposed. Both approaches are used for studying the λ 3,2,1 -numbers of strong, Cartesian and direct products of paths and cycles.

Double Roman domination in trees

Mesh is an important and popular interconnection network topology for large parallel computer systems. A mesh can be divided into submeshes to obtain the upper bounds on the connection probability for the mesh. Combinatorial techniques are used to get closer upper bounds on the connection probability for 2-D meshes compared with the existing upper bounds we have known. Simulation results of meshes of various sizes show that our upper bounds are close to the exact connection probability. The combinatorial methods and tools used in this paper can be used to study the connection probabilities for other networks.

L(3,2,1)-labeling of triangular and toroidal grids

A set D ? V ( G ) of a graph G is a dominating set if every vertex v ? V ( G ) is either in D or adjacent to a vertex in D . The domination number γ ( G ) of a graph G is the minimum cardinality of a dominating set of G . Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n -vertex maximal outerplanar graph having t vertices of degree 2, then γ ( G ) ? n + t 4 . We improve their upper bound by showing γ ( G ) ? n + k 4 , where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ ( G ) ? 5 n 16 for a Hamiltonian maximal planar graph G with n ? 7 vertices.