Vertex Identification in Grids and Prisms

A red-white coloring of a nontrivial connected graph [Formula: see text] of diameter [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], whose [Formula: see text]th coordinate is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text]. It is shown that the grid [Formula: see text] is an ID-graph if and only [Formula: see text] and the prism [Formula: see text] is an ID-graph if and only if [Formula: see text].

Effects of nondimensional distance between two square cylinders on the dissipation characteristics of the complex flow

In this paper, effects of nondimensional distance between two square cylinders on the dissipation characteristics of the complex flow are investigated. The viscosity entropy generation rates around two serial square cylinders and the lift coefficient are analyzed to fully reveal the statistical features of the flow dissipation. Numerical results mainly show that the major viscosity entropy generation rate appears in the shear intersection region of the main flow and local stationary vortex. The viscosity entropy generation rate increases with increasing nondimensional distance ([Formula: see text]). The increasing slope of the viscosity entropy generation rate at a range of [Formula: see text] is greater than that of [Formula: see text]. It is also found that the viscosity entropy generation rate is kept as a constant when the nondimensional distance [Formula: see text] is greater than 5. At [Formula: see text], the effect of downstream square cylinder becomes negligible on the viscosity entropy generation rate. The fluctuating amplitude increases with increasing the nondimensional distance [Formula: see text]. The high-frequency peak is ascribed to the strong vortex shedding around the downstream square cylinder, and the low-frequency peak is ascribed to the weak vortex shedding around the up square cylinder at [Formula: see text]. Although our focus is mainly on the complex flow around two square cylinders, this work demonstrates the viscosity entropy generation rate with increasing nondimensional distance, which provides nice physical insight into understanding the local flow dissipation characteristics around the two serial square cylinders.

Distance Vertex Identification in Graphs

A red-white coloring of a nontrivial connected graph [Formula: see text] is an assignment of red and white colors to the vertices of [Formula: see text] where at least one vertex is colored red. Associated with each vertex [Formula: see text] of [Formula: see text] is a [Formula: see text]-vector, called the code of [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and the [Formula: see text]th coordinate of the code is the number of red vertices at distance [Formula: see text] from [Formula: see text]. A red-white coloring of [Formula: see text] for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of [Formula: see text]. A graph [Formula: see text] possessing an ID-coloring is an ID-graph. The problem of determining those graphs that are ID-graphs is investigated. The minimum number of red vertices among all ID-colorings of an ID-graph [Formula: see text] is the identification number or ID-number of [Formula: see text] and is denoted by [Formula: see text]. It is shown that (1) a nontrivial connected graph [Formula: see text] has ID-number 1 if and only if [Formula: see text] is a path, (2) the path of order 3 is the only connected graph of diameter 2 that is an ID-graph, and (3) every positive integer [Formula: see text] different from 2 can be realized as the ID-number of some connected graph. The identification spectrum of an ID-graph [Formula: see text] is the set of all positive integers [Formula: see text] such that [Formula: see text] has an ID-coloring with exactly [Formula: see text] red vertices. Identification spectra are determined for paths and cycles.

Upper triangle free detour number of a graph

For any two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the [Formula: see text] path [Formula: see text] is called a [Formula: see text] triangle free path if no three vertices of [Formula: see text] induce a triangle. The triangle free detour distance [Formula: see text] is the length of a longest [Formula: see text] triangle free path in [Formula: see text]. A [Formula: see text] path of length [Formula: see text] is called a [Formula: see text] triangle free detour. A set [Formula: see text] is called a triangle free detour set of [Formula: see text] if every vertex of [Formula: see text] lies on a [Formula: see text] triangle free detour joining a pair of vertices of [Formula: see text]. The triangle free detour number [Formula: see text] of [Formula: see text] is the minimum order of its triangle free detour sets and any triangle free detour set of order [Formula: see text] is a triangle free detour basis of [Formula: see text]. A triangle free detour set [Formula: see text] of [Formula: see text] is called a minimal triangle free detour set if no proper subset of [Formula: see text] is a triangle free detour set of [Formula: see text]. The upper triangle free detour number [Formula: see text] of [Formula: see text] is the maximum order of its minimal triangle free detour sets and any minimal triangle free detour set of order [Formula: see text] is an upper triangle free detour basis of [Formula: see text]. We determine bounds for it and characterize graphs which realize these bounds. For any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. Also, for any four positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], it is shown that there exists a connected graph [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is the upper detour number, [Formula: see text] is the upper detour monophonic number and [Formula: see text] is the upper geodetic number of a graph [Formula: see text].

Heegaard distance of the link complements in S3

We show that, for any integers, [Formula: see text] and [Formula: see text], there exists a link in [Formula: see text] such that its complement has a genus [Formula: see text] Heegaard splitting with distance [Formula: see text].

A Thermodynamic Formulation of Home Prices in a Monocentric City

In this work, we proposed a theoretical framework inspired by physical thermodynamics to explain the housing price distributions in monocentric cities. In the same spirit as the Alonso–Muth–Mills (AMM) model, we assume that the disposable income [Formula: see text] after renting a home a distance [Formula: see text] from the center of a city is determined by the wage [Formula: see text] generated at the point-like Central Business District (CBD), the rent [Formula: see text], and the transportation cost [Formula: see text]. Unlike in the AMM model, where the scaling exponents are phenomenological, we admitted only physically reasonable exponents for the scaling of various quantities with distance [Formula: see text] from the CBD. We then determine the equilibrium rent [Formula: see text] by requiring [Formula: see text], where we assumed for simplicity the utility function [Formula: see text] (representing the demand side) has diminishing return in [Formula: see text]. In the simplest model, the equilibrium rent is given by [Formula: see text], i.e., the scaling of [Formula: see text] with [Formula: see text] is entirely determined by [Formula: see text]. We then introduce additional home availability [Formula: see text] (representing the supply side) into the simple theory in the form of an entropic correction, [Formula: see text]. The equilibrium rent then becomes [Formula: see text]. This allows us to treat additional availability due to the two-dimensional nature of cities, as well as that due to high-rise buildings on equal footing. Finally, we compare the equilibrium theory against urban data in Singapore, London and Philadelphia. For Singapore, we find quantitative agreement between theory and data. For London, we find only qualitative agreement between theory and data because the transportation cost is zone based. For Philadelphia, the home price distribution is very different from Singapore and London, and shows clear signs of economic segregation, which is difficult to treat in our equilibrium theory.

The edge signal number of a graph

For two vertices [Formula: see text] and [Formula: see text] in a connected graph [Formula: see text], the signal distance [Formula: see text] from [Formula: see text] to [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is a path connecting [Formula: see text] and [Formula: see text], [Formula: see text] is the length of the path [Formula: see text] and in the sum [Formula: see text] runs over all the internal vertices between [Formula: see text] and [Formula: see text] in the path [Formula: see text]. A path between the vertices [Formula: see text] and [Formula: see text] of length [Formula: see text] is called a [Formula: see text] geosig path. A set [Formula: see text] is called a signal set, if every vertex [Formula: see text] in [Formula: see text] lies on a geosig path joining a pair of vertices of [Formula: see text]. The signal number [Formula: see text] is the minimum order of a signal set of a graph [Formula: see text]. An edge signal cover of [Formula: see text] is a set [Formula: see text] such that every edge of [Formula: see text] is contained in a geosig path joining some pair of vertices of [Formula: see text]. The edge signal number [Formula: see text] of [Formula: see text] is the minimum order of an edge signal cover and any edge signal cover of order [Formula: see text] is an edge signal basis of [Formula: see text]. In this paper, we initiate a study on the edge signal number of a graph [Formula: see text].