Shahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201On a class of skew Dyck paths305319246810.22061/jdma.2025.12170.1141ENYvonne WakuthiiKariukiDepartment of Mathematics, Kibabii University, Bungoma, Kenya.Isaac OwinoOkothDepartment of Pure and Applied Mathematics, School of Mathematics, Statistics and Actuarial Science, Maseno University, Maseno, KenyaJournal Article20250621This paper introduces the set of skew 2-Dyck paths- Dyck-like lattice paths that allow unit up-steps, down-steps of length 2, and left-steps of length 2, provided the paths remain non intersecting. An explicit enumeration formula for these paths is derived using the symbolic method and the Lagrange Inversion Formula. In addition, the paper defines three related combinatorial structures: 2-labeled box paths, 3-leaf-labeled plane trees, and 2-edge-labeled plane trees. Bijections are constructed between the set of skew 2-Dyck paths and the set of each of these three structures, thereby demonstrating their enumerative equivalence.https://jdma.sru.ac.ir/article_2468_aba20bf7021725c5571270ffce6e0b14.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201Path length of protected nodes in random binary search trees321332246710.22061/jdma.2025.12370.1151ENRaminKazemiDepartment of Statistics, Faculty of Science, Imam Khomeini International University, Qazvin, I. R. IranSedighehZamani MehreyanDepartment of Statistics, Faculty of Science, Imam Khomeini International University, Qazvin, I. R. IranJournal Article20250809A protected node is a node that is not a leaf and none of its children is a leaf, and also a weakly protected node is not a leaf and at least one of its children is not a leaf. Let <em>P<sub>n</sub></em> and <em>W<sub>n</sub></em> be the path length of the protected and weakly protected nodes in a random binary search tree (BST) of size <em>n</em>, respectively. In this paper, we derive the exact mean and variance of these random variables and show that 15P<sub>n</sub>/11n.ln n → 1 and 15W<sub>n</sub>/14n.ln n→ 1 in probability.https://jdma.sru.ac.ir/article_2467_697032b3edd94ff5c2c3cf82aa040751.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201VelvetFlow: An engineering pipeline for robust multi-density clustering333358247010.22061/jdma.2025.12039.1131ENHosseinEyvaziDepartment of Computer Science, University of Tarbiat Modares, Tehran, I. R. IranMohammadBadzohrehDepartment of Computer Science, University of Tarbiat Modares, Tehran, I. R. Iran0009-0009-4488-581XSeyed AliShahrokhiDepartment of Computer Science, University of Tarbiat Modares, Tehran, I. R. IranJournal Article20250510<strong>Problem. </strong>Real-world datasets seldom respect a single density scale: tight blobs, elongated ribbons, and isolated points often coexist. Classical algorithms such as DBSCAN or \textit{k}-means require domain-specific parameter tuning and provide only ad-hoc support for anomaly detection.<br /><strong>Solution.</strong> We introduce <em>VelvetFlow</em>, an <em>engineering pipeline</em> that turns a set of well-understood building blocks into a cohesive, end-to-end workflow for multi-density clustering \emph{and} principled outlier detection. The pipeline is composed of three reusable stages:<br />(i) \emph{Contextual-density splitting} assigns every point to a high- or low-density partition using a single neighbourhood size $k$.<br />(ii) \emph{Density-aware clustering} applies a Jaccard-guided \textit{FusedNeighbor}+DBSCAN routine to the sparse partition and HDBSCAN to the dense partition-without introducing new hyper-parameters.<br />(iii) \emph{Scaled-MST verification} re-examines the complete $k$-NN graph, flags weakly connected components, and validates them with a $k$-NN gate; this step recovers small remote clusters while filtering genuine anomalies.https://jdma.sru.ac.ir/article_2470_80079c2c7af005d1422668f75806ee13.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201Detection of communities by modularity matrix359374246910.22061/jdma.2025.11684.1112ENAkramHosseiniDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. IranGholamhoseinFath-TabarDepartment of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. IranHosseinShabaniFaculty of Computer, Network and Communication , Imam Hossein Comprehensive University, Tehran, I. R. IranJournal Article20250126One of the most discussed topics in social networks is community detection. As these networks become more complex, spectral graph properties and graph-related structures are increasingly used for community detection. In this paper, we examine these properties of the modularity matrix, such as the eigenvalues of the modularity matrix structure of some specific graphs, modularity energy, and the Estrada modularity index. Additionally, we study the bounds for the energy and Estrada indices. Furthermore, considering the significant issue of estimating the number of communities in some community detection algorithms in networks, we focus on the modularity eigenvalues.https://jdma.sru.ac.ir/article_2469_5385aa6cb431ed6878702679d6b61d26.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201Gutman index of polyomino chains375392246610.22061/jdma.2025.12559.1168ENLailaAzamiDepartment of Mathematics, Faculty of Basic Sciences, Shahed University, Tehran, I. R. IranNaderJafari RadDepartment of Mathematics, Faculty of Basic Sciences, Shahed University, Tehran, I. R. IranJournal Article20250923The Gutman index is a degree-distance-based topological descriptor of connected graphs. In this paper, we derive explicit analytic expressions for its expected value in polyomino chains built by sequentially attaching square tiles via one of two fixed local connection modes. This expectation is expressed as a cubic polynomial in the number of tiles $n$. We then identify which attachment patterns yield the extremal (maximum and minimum) values and compute the overall average of the Gutman index across all polyomino chains of length $n$. These results enhance the topological analysis of square-tiled networks with applications in chemical graph theory, polymer science, and materials design.https://jdma.sru.ac.ir/article_2466_84ade09965346b007e7601ab9e0fbed2.pdfShahid Rajaee Teacher Training UniversityJournal of Discrete Mathematics and Its Applications2981-080910420251201On the characterization of tricyclic graphs with Szeged complexity one393401246510.22061/jdma.2025.12502.1161ENZahraVaziriDepartment of Mathematics, Factually of Science, Shahid Rajaee Teacher Training University, Tehran, I. R. IranJournal Article20250913This paper presents a classification of 12 out of 15 known families of tricyclic graphs based on their Szeged complexity. It is shown that only two of these families contain graphs with Szeged complexity equal to one. Building on previous structural analyses of unicyclic and bicyclic graphs, this study extends the classification framework to include a substantial portion of tricyclic configurations. The results contribute to a deeper understanding of graph complexity and lay the groundwork for further exploration of cyclic graph structures.https://jdma.sru.ac.ir/article_2465_d97b1bbe47ca514983f3998d417eadca.pdf