Search Results (16)

As the global energy transition accelerates, the integration of solar power and artificial intelligence (AI) technologies offers new pathways for sustainable local development. This study examines four Serbian municipalities—Šabac, Sombor, Pirot, and Čačak—to assess how AI-enabled solar power systems can enhance energy resilience, reduce emissions, and support community-level sustainability goals. Using a mixed-method approach combining spatial analysis, predictive modeling, and stakeholder interviews, this research study evaluates the performance and institutional readiness of local governments in terms of implementing intelligent solar infrastructure. Key AI applications included solar potential mapping, demand-side management, and predictive maintenance of photovoltaic (PV) systems. Quantitative results show an improvement >60% in forecasting accuracy, a 64% reduction in system downtime, and a 9.7% increase in energy cost savings. These technical gains were accompanied by positive trends in SDG-aligned indicators, such as improved electricity access and local job creation in the green economy. Despite challenges related to data infrastructure, regulatory gaps, and limited AI literacy, this study finds that institutional coordination and leadership commitment are decisive for successful implementation. The proposed AI–Solar Integration for Local Sustainability (AISILS) framework offers a replicable model for emerging economies. Policy recommendations include investing in foundational digital infrastructure, promoting low-code AI platforms, and aligning AI–solar projects with SDG targets to attract EU and national funding. This study contributes new empirical evidence on the digital–renewable energy nexus in Southeast Europe and underscores the strategic role of AI in accelerating inclusive, data-driven energy transitions at the municipal level. Full article

The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the idea of optimization in the families of bicyclic graphs, trees, and unicyclic graphs. We investigated optimal values in the stated families by means of well-known transformations. The transformations include the following: Transformation A, Transformation B, Transformation C, and Transformation D. Transformation A and Transformation B increase the value of the generalized Sombor index, while Transformation C and Transformation D are used for minimal values. Full article

Fractals are geometric patterns that appear self-similar across all length scales and are constructed by repeating a single unit on a regular basis. Entropy, as a core thermodynamic function, is an extension based on information theory (such as Shannon entropy) that is used to describe the topological structural complexity or degree of disorder in networks. Topological indices, as graph invariants, provide quantitative descriptors for characterizing global structural properties. In this paper, we investigate two types of generalized Sierpiński graphs constructed on the basis of different seed graphs, and employ six topological indices—the first Zagreb index, the second Zagreb index, the forgotten index, the augmented Zagreb index, the Sombor index, and the elliptic Sombor index—to analyze the corresponding entropy. We utilize the method of edge partition based on vertex degrees and derive analytical formulations for the first Zagreb entropy, the second Zagreb entropy, the forgotten entropy, the augmented Zagreb entropy, the Sombor entropy, and the elliptic Sombor entropy. This research approach, which integrates entropy with Sierpiński network characteristics, furnishes novel perspectives and instrumental tools for addressing challenges in chemical graph theory, computer networks, and other related fields. Full article

In this paper, motivated by the recently introduced topological indices—the Sombor index, Sombor coindex, and Lanzhou index, we define a new index—the Lanzhou coindex of a graph. Furthermore, we investigate the Sombor index (coindex) and the Lanzhou index (coindex) of tadpole graphs, wheel graphs, and two-dimensional grid graphs, as well as their paraline graphs. Full article

A connected graph with p vertices and q edges satisfying $q=p+1$ is referred to as a bicyclic graph. This paper is concerned with an optimal study of the BID (bond incident degree) indices of fixed-size bicyclic graphs with a given matching number. Here, a BID index of a graph G is the number ${\mathcal{BID}}_{\mathfrak{f}}\left(G\right)={\sum }_{vw\in E\left(G\right)}\mathfrak{f}(d_G\left(v\right),d_G\left(w\right))$, where $E\left(G\right)$ represents G’s edge set, $d_G\left(v\right)$ denotes vertex v’s degree, and $\mathfrak{f}$ is a real-valued symmetric function defined on the Cartesian square of the set of all different members of G’s degree sequence. Our results cover several existing indices, including the Sombor index and symmetric division deg index. Full article

Consider a unicyclic graph G with edge set $E\left(G\right)$. Let $\mathfrak{f}$ be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)={\sum }_{xy\in E\left(G\right)}\mathfrak{f}(d_G\left(x\right),d_G\left(y\right))$, where $d_G\left(x\right)$ denotes the degree a vertex x in G. This paper determines optimal bounds for ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)$ in terms of the order of G and a parameter $\mathfrak{z}$, where $\mathfrak{z}$ is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function $\mathfrak{f}$ must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices. Full article

The inner core of Sombor, known as “Venac”, is probably the best-preserved one among medium-sized cities in Serbia. The stagnation of Sombor during the 20th century and its urban shrinkage in the 21st century have prevented significant transformations of the core, enabling its preservation under state protection as an urban heritage site. However, the recent rise of cultural tourism has triggered urban regeneration. As the city is still unprepared for this change, this regeneration has mostly omitted the inner core. Realising this, local representatives and experts have started rethinking innovative approaches to its regeneration, including the concept of Albergo Diffuso. This sustainable concept is created to revive the historic cores of small, shrinking cities and towns. Basically, it represents a hotel situated in several old buildings dispersed throughout a historic urban fabric, fitting perfectly into the regeneration of Venac. However, the current lack of precise spatial indicators and thresholds makes their incorporation into the planning process challenging. Considering this, this study focuses on the current spatial development of tourism in Venac, analysing the elements that would support and facilitate the application of this concept in the future. This article also proposes a set of new planning measures to support a strategically organised approach—from the emphasis on urban reuse and physical renewal to multileveled linking of basic concept conditions to the prioritization of pedestrian-friendly places and the application of innovative urban design in open public spaces. By connecting the selected Albergo Diffuso approach with spatial development and its analysis, this study also contributes to the spatial imprint of the concept’s implementation. Full article

A vertex degree based topological index called the Sombor index was recently defined in 2021 by Gutman and has been very popular amongst chemists and mathematicians. We determine the amount of change of the Sombor index when some elements are removed from a graph. This is done for several graph elements, including a vertex, an edge, a cut vertex, a pendant edge, a pendant path, and a bridge in a simple graph. Also, pendant and non-pendant cases are studied. Using the obtained formulae successively, one can find the Sombor index of a large graph by means of the Sombor indices of smaller graphs that are just graphs obtained after removal of some vertices or edges. Sometimes, using iteration, one can manage to obtain a property of a really large graph in terms of the same property of many other subgraphs. Here, the calculations are made for a pendant and non-pendant vertex, a pendant and non-pendant edge, a pendant path, a bridge, a bridge path from a simple graph, and, finally, for a loop and a multiple edge from a non-simple graph. Using these results, the Sombor index of cyclic graphs and tadpole graphs are obtained. Finally, some Nordhaus–Gaddum type results are obtained for the Sombor index. Full article

The cloud computing networks used in the IoT, and other themes of network architectures, can be investigated and improved by cheminformatics, which is a combination of chemistry, computer science, and mathematics. Cheminformatics involves graph theory and its tools. Any number that can be uniquely calculated by a graph is known as a graph invariant. In graph theory, networks are converted into graphs with workstations or routers or nodes as vertex and paths, or connections as edges. Many topological indices have been developed for the determination of the physical properties of networks involved in cloud computing. The study computed newly prepared topological invariants, K-Banhatti Sombor invariants (KBSO), Dharwad invariants, Quadratic-Contraharmonic invariants (QCI), and their reduced forms with other forms of cloud computing networks. These are used to explore and enhance their characteristics, such as scalability, efficiency, higher throughput, reduced latency, and best-fit topology. These attributes depend on the topology of the cloud, where different nodes, paths, and clouds are to be attached to achieve the best of the attributes mentioned before. The study only deals with a single parameter, which is a topology of the cloud network. The improvement of the topology improves the other characteristics as well, which is the main objective of this study. Its prime objective is to develop formulas so that it can check the topology and performance of certain cloud networks without doing or performing experiments, and also before developing them. The calculated results are valuable and helpful in understanding the deep physical behavior of the cloud’s networks. These results will also be useful for researchers to understand how these networks can be constructed and improved with different physical characteristics for enhanced versions. Full article

Climate factors have an impact on plant life cycle, yield, productivity, economy and profitability of agricultural production. There are not a lot of studies on understanding of influence of climate factors variation on maize yield in agro-ecological conditions of Serbia. The aim of this paper is analysis of variation of climatic factors over a long-time period, as well as assessment of impact of the examined climate parameters on maize yield in two localities in the Republic of Serbia. For the analysis of climatic factors (temperature, precipitation, sunshine, humidity) in the region of Central Serbia and Vojvodina, the data of meteorological stations Kragujevac and Sombor during two thirty-year periods (1961–1990 and 1991–2020) were used. In order to determine the existence and strength of the relationship between the observed climatic factors and maize yield, a correlation analysis of these indicators for the period 2005–2020 years, was performed. In the period 1991–2020, the average values of temperature were annually increased for 0.046 °C in Kragujevac and for 0.05 °C in Sombor, and in the same period the average value of sunshine on an annual level was increased for 1.3 h in Kragujevac and for 5.01 h in Sombor, 2020 in comparison to average values in period of 1961–1990. The humidity was decline annually for 1.3 in Kragujevac and for 3.4 in Sombor in period 1991–2020 in comparison to average humidity in period of 1961–1990. The results of the correlation analysis showed that the maize yield was significantly lower in the years with expressed high temperatures and precipitation deficit. Based on these studies, established effect of climate change on maize yield and that this demand developing adaptation agricultural practice through creating maize hybrids and varieties with greater adaptability and improvement of agrotechnic measure. Full article

Let G be a simple graph with the vertex set $V=\{v_1,\dots ,v_n\}$ and denote by $d_{v_i}$ the degree of the vertex $v_i$. The modified Sombor index of G is the addition of the numbers ${(d_{v_i}^2+d_{v_j}^2)}^{-1/2}$ over all of the edges $v_i{v}_j$ of G. The modified Sombor matrix $A_{MS}\left(G\right)$ of G is the n by n matrix such that its $(i,j)$-entry is equal to ${(d_{v_i}^2+d_{v_j}^2)}^{-1/2}$ when $v_i$ and $v_j$ are adjacent and 0 otherwise. The modified Sombor spectral radius of G is the largest number among all of the eigenvalues of $A_{MS}\left(G\right)$. The sum of the absolute eigenvalues of $A_{MS}\left(G\right)$ is known as the modified Sombor energy of G. Two graphs with the same modified Sombor energy are referred to as modified Sombor equienergetic graphs. In this article, several bounds for the modified Sombor index, the modified Sombor spectral radius, and the modified Sombor energy are found, and the corresponding extremal graphs are characterized. By using computer programs (Mathematica and AutographiX), it is found that there exists only one pair of the modified Sombor equienergetic chemical graphs of an order of at most seven. It is proven that the modified Sombor energy of every regular, complete multipartite graph is $\sqrt{2}$; this result gives a large class of the modified Sombor equienergetic graphs. The (linear, logarithmic, and quadratic) regression analyses of the modified Sombor index and the modified Sombor energy together with their classical versions are also performed for the boiling points of the chemical graphs of an order of at most seven. Full article

In this paper, we introduce some new versions based on the locating vectors named locating indices. In particular, Hyper locating indices, Randić locating index, and Sambor locating index. The exact formulae for these indices of some well-known families of graphs and for the Helm graph are derived. Moreover, we determine the importance of these locating indices for 11 benzenoid hydrocarbons. Furthermore, we show that these new versions of locating indices have a reasonable correlation using linear regression with physicochemical characteristics such as molar entropy, acentric factor, boiling point, complexity, octanol–water partition coefficient, and Kovats retention index. The cases in which good correlations were obtained suggested the validity of the calculated topological indices to be further used to predict the physicochemical properties of much more complicated chemical compounds. Full article

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Then, the Sombor index of G is defined as $SO\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$. In this paper, we give some lemmas that can be used to compare the Sombor indices between two graphs. With these lemmas, we determine the graph with maximum $SO$ among all cacti with n vertices and k cut edges. Furthermore, the unique graph with maximum $SO$ among all cacti with n vertices and p pendant vertices is characterized. In addition, we find the extremal graphs with respect to $SO$ among all quasi-unicyclic graphs. Full article

Let G be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. A graph invariant for G is a number related to the structure of G which is invariant under the symmetry of G. The Sombor and reduced Sombor indices of G are two new graph invariants defined as $SO\left(G\right)={\sum }_{uv\in E\left(G\right)}\sqrt{d_G{\left(u\right)}^2+d_G{\left(v\right)}^2}$ and $S{O}_{red}\left(G\right)={\sum }_{uv\in E\left(G\right)}\sqrt{{\left(d_G\left(u\right)-1\right)}^2+{\left(d_G\left(v\right)-1\right)}^2}$, respectively, where $d_G\left(v\right)$ is the degree of the vertex v in G. We denote by $H_{n,\nu }$ the graph constructed from the star $S_n$ by adding $\nu$ edge(s), $0\le \nu \le n-2$, between a fixed pendent vertex and $\nu$ other pendent vertices. Réti et al. [T. Réti, T Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math.3 (2021) 11–18] proposed a conjecture that the graph $H_{n,\nu }$ has the maximum Sombor index among all connected $\nu$-cyclic graphs of order n, where $0\le \nu \le n-2$. In some earlier works, the validity of this conjecture was proved for $\nu \le 5$. In this paper, we confirm that this conjecture is true, when $\nu =6$. The Sombor index in the case that the number of pendent vertices is less than or equal to $n-\nu -2$ is investigated, and the same results are obtained for the reduced Sombor index. Some relationships between Sombor, reduced Sombor, and first Zagreb indices of graphs are also obtained. Full article

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix. Full article