https://jdma.sru.ac.ir/ Journal of Discrete Mathematics and Its Applications en daily 1 Sun, 01 Mar 2026 00:00:00 +0330 Sun, 01 Mar 2026 00:00:00 +0330 https://jdma.sru.ac.ir/article_2364.html Let $G=(V,E)$ be a simple graph and $f$ a function defined from $V$ to $\{0,1,2,3\}.$ A vertex $u$ with $f(u)=0$ is called an undefended vertex with respect to $f$ if it is not adjacent to a vertex $v$ with $f(v)\geq2$. The function $f$ is called a generous Roman dominating function (GRD-function) if for every vertex with $f(u)=0$ there exists at least a vertex $v$ with $f(v)\geq2$ adjacent to $u$ such that the function $f^{\prime}:V\rightarrow\{0,1,2,3\}$, defined by $f^{\prime}(u)=\alpha$, $f^{\prime}(v)=f(v)-\alpha$ where $\alpha\in\{1,2\}$, and $f^{\prime}(w)=f(w)$ if $w\in V-\{u,v\}$ has no undefended vertex. The weight of a GRD-function $f$ is the sum of its function values over all vertices, and the minimum weight of a GRD-function on $G$ is the generous Roman domination number $\gamma_{gR}(G)$. The $\gamma_{gR}$-stability $\mathrm{st}_{\gamma_{gR}}(G)$ (resp. $\gamma_{gR}^{-}$-stability $\mathrm{st}_{\gamma_{gR}}^{-}(G)$, $\gamma_{gR}^{+}$-stability $\mathrm{st}_{\gamma_{gR}}^{+}(G)$) of $G$ is defined as the order of the smallest set of vertices whose removal changes (resp. decreases, increases) the generous Roman domination number. In this paper, we first determine the exact values of $\gamma_{gR}$-stability for some special classes of graphs, and then we present some bounds on $\mathrm{st}_{\gamma_{gR}}(G)$. We also characterize graphs with large $\mathrm{st}_{\gamma_{gR}}(G)$.Moreover, we show that if $T$ is a nontrivial tree, then $\mathrm{st}_{\gamma_{gR}}(T)\leq2,$ and if further $T$ has maximum degree $\Delta\geq3$, then $\mathrm{st}_{\gamma_{gR}}^{-}(T)\leq\Delta-1$. https://jdma.sru.ac.ir/article_2547.html This study explores a novel authentication algorithm leveraging ECG signals and deep learning models, specifically VGG16, enhanced with transfer learning. Authentication systems were evaluated based on preparation time, response time, and accuracy, with biometric data utilized to increase security. Traditional deep learning models face challenges in retraining time when data changes, prompting the proposed algorithm to incorporate new users or modify access efficiently using transfer learning. Key findings included training the VGG16 model on ECG data from 48 individuals (MITDB dataset) with a 99.45% accuracy and 120 seconds average training time. The transfer learning approach enabled adding or removing user data by adjusting SoftMax coefficients, reducing training time significantly to about 12 seconds per user with accuracy exceeding 99%. Removing a user followed a similar process with comparable results. Overall, the algorithm reduced retraining time by 72.89% while maintaining over 99.16% accuracy. Additionally, the system's response time for a new user was 37 milliseconds, demonstrating practicality for real-time applications. The study highlights the proposed algorithm's efficiency in managing user data changes while ensuring high accuracy and reduced retraining time, making it a robust solution for modern authentication systems. https://jdma.sru.ac.ir/article_2494.html Generalized stepwise irregular (GSI) graphs are graphs in which the degree difference between every pair of adjacent vertices is positive constant. Specifically, a graph $G $ is called a $ k $-stepwise irregular ( k -SI) graph if $|d_G(u)-d_G(v)|=k$ for each edge $uv \in E(G) $. In this paper, We examine the behavior of GSI graphs under some graph operations, such as sum, corona product, complement, subdivision, line graph, and vertex deletion. An Infinite family of $ 3$-SI graphs with a given cyclomatic number and distinct cycles are constructed. Further, a lower bound on the size of the unicyclic $3$-SI graphs is proposed. https://jdma.sru.ac.ir/article_2548.html The elliptic Sombor index serves as a topological index that is based on vertex degrees. Our research focuses on the elliptic Sombor index within various graph types, including $m^{k}$-graphs, Jahangir graphs, Barbell graphs, and Friendship graphs. Additionally, we derive this index for the subdivided $m^{k}$-graphs. Furthermore, we present several u bounds for some of these indices. https://jdma.sru.ac.ir/article_2549.html In this paper, we study the enumeration of vertices in $d$-dimensional plane trees with respect to their levels and degrees. This class of trees generalizes both ordinary plane trees and noncrossing trees. Our approach builds upon a decomposition framework that extends the butterfly decomposition of plane trees introduced by Chen, Li and Shapiro, as well as that of noncrossing trees studied by Oduol and Okoth. We derive both explicit and asymptotic formulas for the enumeration of vertices, eldest children, first children, non-first children and non-leaves at specified levels and degrees. The results are obtained through a combination of generating function techniques, refined butterfly decompositions and bijective methods. This work extends previous enumeration results on ordinary plane trees and noncrossing trees and provides new insights into the combinatorial structure of their higher-dimensional analogues. https://jdma.sru.ac.ir/article_2550.html For a finite group G, the prime order element graph of G, is defined as follows: the vertex set is the elements of G and two distinct vertices a and b are adjacent if and only if order of ab is prime. This graph is denoted by \Gamma(G). In this paper, we prove that the alternating groups A_4 and A_5, all groups of order pq, where p and q are prime numbers, and cyclic groups Z_p of prime order are uniquely characterized by the prime order element graph.