In this paper, the weighted Szeged indices of Cartesian product and Corona product of two connected graphs are obtained. Using the results obtained here, the weighted Szeged indices of the hypercube of dimension n, Hamming graph, C4 nanotubes, nanotorus, grid, t-fold bristled, sunlet, fan, wheel, bottleneck graphs and some classes of bridge graphs are computed.

This paper represents a technique for finding optimal paths with multiple indexes in a graph. Up to the present time, all optimal paths have been determined upon one index, say, distance for which an evaluation method exists. In this paper firstly we define multiple indexes for each edge in such a way that anyone can treat the factor for assigning an optimal path. Here, we use Data Envelopment Analysis (DEA) technique for designing a model that can identify optimal paths with multiple indexes, and separate them from the other paths.

Let $G$ be a graph with a vertex weight $omega$ and the vertices $v_1,ldots,v_n$. The Laplacian matrix of $G$ with respect to $omega$ is defined as $L_omega(G)=diag(omega(v_1),cdots,omega(v_n))-A(G)$, where $A(G)$ is the adjacency matrix of $G$. Let $mu_1,cdots,mu_n$ be eigenvalues of $L_omega(G)$. Then the Laplacian energy of $G$ with respect to $omega$ defined as $LE_omega (G)=sum_{i=1}^nbig|mu_i - overline{omega}big|$, where $overline{omega}$ is the average of $omega$, i.e., $overline{omega}=dfrac{sum_{i=1}^{n}omega(v_i)}{n}$. In this paper we consider several natural vertex weights of $G$ and obtain some inequalities between the ordinary and Laplacian energies of $G$ with corresponding vertex weights. Finally, we apply our results to the molecular graph of toroidal fullerenes (or achiral polyhex nanotorus).\[5mm] noindenttextbf{Key words:} Energy of graph, Laplacian energy, Vertex weight, Topological index, toroidal fullerenes.

Nowadays, whereas the use of social networks and computer networks is increasing, the amount of associated complex data with graph structure and their applications, such as classification, clustering, link prediction, and recommender systems, has risen significantly. Because of security problems and societal concerns, anomaly detection is becoming a vital problem in most fields. Applications that use a heterogeneous graph, are confronted with many issues, such as different kinds of neighbors, different feature types, and differences in type and number of links. So, in this research, we employ the HetGNN model with some changes in loss functions and parameters for heterogeneous graph embedding to capture the whole graph features (structure and content) for anomaly detection, then pass it to a VAE to discover anomalous nodes based on reconstruction error. Our experiments on AMiner data set with many base-lines illustrate that our model outperforms state-of-the-arts methods in heterogeneous graphs while considering all types of attributes.

The Bertz indices, derived by counting the number of connecting edges of line graphs of a molecule were used in deriving the QSPR models for the physicochemical properties of alkanes. The inability of these indices to identify the hetero centre in a chemical compound restricted their applications to hydrocarbons only. In the present work, a novel molecular descriptor has been derived from the weighted line graph of the molecular structure and applied in correlating the physicochemical properties of alkane isomers with these descriptors. A weight is tagged at the vertex of the line graph, which consequently modifies the weight of the edge. These descriptors were found to classify the alkane isomers and served well in deriving the QSPR models for various physicochemical properties. The mathematical calculations include the quantitative treatment on the role of substituents (alkyl) in governing the properties under study of the alkane isomers. Further, the use of weighted line graph in the enumeration of the topological index opens up a new vista on application to heteroatomic systems.

Feature extraction plays a crucial role in dimensionality reduction in machine learning applications. Nonnegative Matrix Factorization (NMF) has emerged as a powerful technique for dimensionality reduction; however, its equal treatment of all features may limit accuracy. To address this challenge, this paper introduces Graph-Regularized Entropy-Weighted Nonnegative Matrix Factorization (GEWNMF) for enhanced feature representation. The proposed method improves feature extraction through two key innovations: optimizable feature weights and graph regularization. GEWNMF uses optimizable weights to prioritize the extraction of crucial features that best describe the underlying data structure. These weights, determined using entropy measures, ensure a diverse selection of features, thereby enhancing the fidelity of the data representation. This adaptive weighting not only improves interpretability but also strengthens the model against noisy or outlier-prone datasets. Furthermore, GEWNMF integrates robust graph regularization techniques to preserve local data relationships. By constructing an adjacency graph that captures these relationships, the method enhances its ability to discern meaningful patterns amid noise and variability. This regularization not only stabilizes the method but also ensures that nearby data points appropriately influence feature extraction. Thus, GEWNMF produces representations that capture both global trends and local nuances, making it applicable across various domains. Extensive experiments on four widely used datasets validate the efficacy of GEWNMF compared to existing methods, demonstrating its superior performance in capturing meaningful data patterns and enhancing interpretability.

There are various engineering applications dealing with the prototype problem of finding the best p-medians in a weighted graph. However, the heuristic developments are still of concern due to their complexity. This paper utilizes genetic algorithm as a well-known reliable evolutionary search for such a purpose. Problem formulation is studied, introducing a characteristic graph and specialized genotype representation called\Direct Index Coding". The genetic operators are also modified due to problem requirements, and further tuned using a simulated annealing approach. Such an enhanced evolutionary search tool is then applied to a number of examples to show its effectiveness regarding the exact results, and to compare efficiency between tuned and non-tuned GA.

Massive MIMO cellular networks, despite their ability to serve multiple users simultaneously, face a significant challenge due to pilot contamination. This paper presents an innovative two-stage algorithm to reduce this contamination and increase user data rates in both uplink and downlink. The key innovation of the proposed method lies in the intelligent integration of three techniques: Soft Pilot Reuse (SPR), optimal pilot sequence selection algorithm, and Weighted Graph Coloring (WGC). This combination simultaneously addresses three fundamental issues: contamination in SPR due to fixed thresholds, increased pilot overhead in WGC, and the problem of maximizing the data rate of the user with the lowest rate. The proposed algorithm operates in two stages. In the first stage, users are divided into center and edge groups, and optimal pilot sequences are determined based on their data rates. In the second stage, using the WGC algorithm and creating an Edge-Weighted Interference Graph (EWIG), pilot contamination is reduced based on the intensity of user interference. Simulation results show that this method significantly improves system performance compared to the best existing method (WGC). In the downlink, an 11 dB improvement in Signal-to-Interference-plus-Noise Ratio (SINR) and a 0.16 bps/Hz increase in average achievable rate are observed. In the uplink, a 2.4 dB improvement in SINR and a 0.46 bps/Hz increase in average achievable rate are achieved. In terms of computational complexity, the proposed method has lower complexity compared to the WGC scheme. Moreover, energy efficiency analysis confirms the superiority of the proposed method in both uplink and downlink scenarios.