Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices

We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.

Classification of Upper Bound Sequences of Local Fractional Metric Dimension of Rotationally Symmetric Hexagonal Planar Networks

The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.

Improved Networks Routing Using an Arrow-Based Description

In this paper, an improved routing algorithm suitable for planar networks—static Zigbee and mesh networks included—is shown. The algorithm is based on the cycle description of the graph, and on a new graph model based on arrow description, which is outlined. We show that the newly developed model allows for a faster algorithm for finding a direct and a return path in the network. The newly developed model allows further interpretations of the relationships in any simple planar graphs. Examples showing the implementation of the newly developed model are presented too.

Optimal Control Strategy to Distribute Water Through Loop-Like Planar Networks

In this article, loop like planar networks formed by circular cross sectioned conduits with possibly different geometric measurements are studied to supply the required amount of isothermal water within the optimal time and through the shortest path. The flow optimization procedure is controlled by time varying pressures at nodes throughout the network for given specifications about pressure value at multiple demanding and single supply nodes. The flow governing equation is solved analytically to correlate transient flow rate and pressure and then studied using analogous electrical circuit. For each possible path between source and demand node, minimum equivalent flow impedance criterion is considered to pick the optimum path. This sets a multi-objective dynamic flow optimization algorithm and the same is executed under the assumption of fully developed and laminar flow. The optimum flow impedance can further be used to measure the pumping power as the cost of flow of a particular path. The algorithm can be extended to reduce the water wastages by controlling pressures efficiently.