Linear Algorithms for the Hosoya Index and Hosoya Matrix of a Tree

The Hosoya index of a graph is defined as the total number of its independent edge sets. This index is an important example of topological indices, a molecular-graph based structure descriptor that is of significant interest in combinatorial chemistry. The Hosoya index inspires the introduction of a matrix associated with a molecular acyclic graph called the Hosoya matrix. We propose a simple linear-time algorithm, which does not require pre-processing, to compute the Hosoya index of an arbitrary tree. A similar approach allows us to show that the Hosoya matrix can be computed in constant time per entry of the matrix.

Inference of a universal social scale and segregation measures using social connectivity kernels

How people connect with one another is a fundamental question in the social sciences, and the resulting social networks can have a profound impact on our daily lives. Blau offered a powerful explanation: people connect with one another based on their positions in a social space. Yet a principled measure of social distance, allowing comparison within and between societies, remains elusive. We use the connectivity kernel of conditionally independent edge models to develop a family of segregation statistics with desirable properties: they offer an intuitive and universal characteristic scale on social space (facilitating comparison across datasets and societies), are applicable to multivariate and mixed node attributes, and capture segregation at the level of individuals, pairs of individuals and society as a whole. We show that the segregation statistics can induce a metric on Blau space (a space spanned by the attributes of the members of society) and provide maps of two societies. Under a Bayesian paradigm, we infer the parameters of the connectivity kernel from 11 ego-network datasets collected in four surveys in the UK and USA. The importance of different dimensions of Blau space is similar across time and location, suggesting a macroscopically stable social fabric. Physical separation and age differences have the most significant impact on segregation within friendship networks with implications for intergenerational mixing and isolation in later stages of life.

Inference of a universal social scale and segregation measures using social connectivity kernels

How people connect with one another is a fundamental question in the social sciences, and the resulting social networks can have a profound impact on our daily lives. Blau offered a powerful explanation: people connect with one another based on their positions in a social space. Yet a principled measure of social distance, allowing comparison within and between societies, remains elusive. We use the connectivity kernel of conditionally independent edge models to develop a family of segregation statistics with desirable properties: they offer an intuitive and universal characteristic scale on social space (facilitating comparison across datasets and societies), are applicable to multivariate and mixed node attributes, and capture segregation at the level of individuals, pairs of individuals and society as a whole. We show that the segregation statistics can induce a metric on Blau space (a space spanned by the attributes of the members of society) and provide maps of two societies. Under a Bayesian paradigm, we infer the parameters of the connectivity kernel from 11 ego-network datasets collected in four surveys in the UK and USA. The importance of different dimensions of Blau space is similar across time and location, suggesting a macroscopically stable social fabric. Physical separation and age differences have the most significant impact on segregation within friendship networks with implications for intergenerational mixing and isolation in later stages of life.

Aesthetic Perception of Line Patterns: Effect of Edge-Orientation Entropy and Curvilinear Shape

Curvilinearity is a perceptual feature that robustly predicts preference ratings for a variety of visual stimuli. The predictive effect of curved/angular shape overlaps, to a large degree, with regularities in second-order edge-orientation entropy, which captures how independent edge orientations are distributed across an image. For some complex line patterns, edge-orientation entropy is actually a better predictor for what human observers like than curved/angular shape. The present work was designed to disentangle the role of the two features in artificial patterns that consisted of either curved or angular line elements. We systematically varied these patterns across two more dimensions, edge-orientation entropy and the number of lines. Eighty-three participants rated the stimuli along three aesthetic dimensions ( pleasing, harmonious, and complex). Results showed that curved/angular shape was a stronger predictor for ratings of pleasing and harmonious if the stimuli consisted of a few lines that were clearly discernible. By contrast, edge-orientation entropy was a stronger predictor for the ratings if the stimuli showed many lines, which merged into a texture. No such differences were obtained for complexity ratings. Our findings are in line with results from neurophysiological studies that the processing of shape and texture, respectively, is mediated by different cortical mechanisms.

Two Classes of Topological Indices of Phenylene Molecule Graphs

A phenylene is a conjugated hydrocarbons molecule composed of six- and four-membered rings. The matching energy of a graphGis equal to the sum of the absolute values of the zeros of the matching polynomial ofG, while the Hosoya index is defined as the total number of the independent edge sets ofG. In this paper, we determine the extremal graph with respect to the matching energy and Hosoya index for all phenylene chains.

Hosoya Index ofL-Type Polyphenyl Spiders

The polyphenyl system is composed ofnhexagons obtained from two adjacent hexagons that are sticked by a path with two vertices. The Hosoya index of a graphGis defined as the total number of the independent edge sets ofG. In this paper, we give a computing formula of Hosoya index of a type of polyphenyl system. Furthermore, we characterize the extremal Hosoya index of the type of polyphenyl system.