Search Results (8)

Degree-based topological indices play a central role in characterizing graph structures and their chemical applications. Among these, multiplicative Zagreb indices have attracted considerable attention due to their strong discriminative power and relevance in chemical graph theory. Neighborhood versions of these indices extend the classical concept by incorporating the aggregate degree information of adjacent vertices, capturing more subtle structural effects related to local branching. Trees, as connected acyclic graphs, provide a natural and tractable class for studying the extremal behaviors of these indices, while molecular trees—trees with a maximum degree of at most four—serve as chemically meaningful models of acyclic organic compounds. Investigating extremal values on these structures offers both theoretical insight into the indices’ behavior and identification of molecular graphs that maximize or minimize them. In this work, we determine the maximal and minimal values of the neighborhood-based multiplicative Zagreb indices for trees of fixed order and prescribed maximum degree, and we provide a complete structural characterization of all extremal graphs. Special attention is given to molecular trees, for which explicit extremal bounds are derived and all optimal structures are identified. These results provide efficient tools for evaluating the indices and illuminate the structural principles governing their extremal behavior. Full article

This study analyses the influence of three-dimensional (3D) urban form on intra-urban thermal variability and its long-term evolution in Zagreb, Croatia. The research focuses on four residential districts (Špansko sjever, Dugave, Lanište, and Novi Jelkovec) representing different development periods. The central hypothesis is that differences in the development period and urban compactness are associated with differences in summer thermal patterns, with more open spatial configurations generally exhibiting weaker thermal responses than more compact developments. The methodology integrates LiDAR-derived building morphology with a decade-long Landsat time series (2015–2024), including land surface temperature (LST), normalized difference vegetation index (NDVI), and normalized difference built-up index (NDBI). The results indicate a consistent increase in summer LST across all analysed neighbourhoods, with warming rates ranging from approximately 2.00 to 2.83 °C per decade. Built-up intensity shows a positive association with temperature, while vegetation trends are generally weak. A multiple linear regression model explains 47% of the variance in LST (R² = 0.47), with NDBI identified as a significant predictor (p < 0.01), whereas NDVI and volumetric building density are not statistically significant. Despite this, neighbourhoods with higher volumetric building density (up to ≈2.96 m³/m²) tend to exhibit stronger warming trends compared to lower-density areas (≈1.69 m³/m²), indicating the additional explanatory value of three-dimensional urban morphology. These findings support the concept of a volumetric expression of urban thermal processes, while highlighting that 3D urban morphology contributes to the interpretation of the long-term thermal patterns when considered alongside other factors. They also emphasize the importance of integrating 3D spatial metrics into climate-sensitive urban planning and mitigation strategies. Full article

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for $\alpha \in {\mathbb{R}}^{+}\backslash \left\{1\right\}.$ Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for $\alpha \in {\mathbb{R}}^{+}\backslash \left\{1\right\},$ we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. Full article

Urban streams constitute a valuable form of multi-functional blue and green infrastructure (BGI) and can support urban development to generate ecosystem, social, and economic benefits. In global cities, planning for BGI enhancement contributes to climate change adaptation, ecosystem restoration, community health and wellbeing, improved quality of life, etc. This research aims to assess the dynamics of stream landscape change in Zagreb as well as the influence of urban development on the blue and green landscape and related urban values. The analysis of landscape features and their planning is conducted at the level of the stream system of the whole city of Zagreb and at the level of two stream sequences by superimposing spatial data from cartographic sources. By developing an urban planning-social-ecological approach to evaluation, monitoring, and management, a quantitative and qualitative trend in stream landscape changes is identified and indicators for detecting areas of critical urbanization pressure are established. This research confirms the trend of negative changes in the urban BGI, evident in the present state (interruption of open streams, and the reduction, fragmentation, and disconnection of blue and green landscape), as well as in the planned neglect (plans for further stream closures and reduction in public green areas). Stream landscape potential is emphasized as one of the greatest urban assets for improving the system of BGI, and areas needed for their prioritization in urban planning measures, directed towards an increase in multiple landscape values, are determined. Full article

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as “subdivision vertex-edge join” of three graphs. Full article

We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results. Full article

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon $S{i}_2{C}_{3^{-}}I\left[p,q\right]$ and $S{i}_2{C}_{3^{-}}\mathrm{II}\left[p,q\right]$ second. Full article

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this report, we compute closed forms of first Zagreb, second Zagreb, and forgotten polynomials of generalized prism and toroidal polyhex networks. We also compute hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and forgotten index of these networks. Moreover we gave graphical representation of our results, showing the technical dependence of each topological index and polynomial on the involved structural parameters. Full article