Let [Formula: see text] be a connected graph, and let [Formula: see text] and [Formula: see text] be two positive integers with [Formula: see text] (mod 2). A [Formula: see text]-odd factor of [Formula: see text] is a spanning subgraph [Formula: see text] of [Formula: see text] with [Formula: see text] (mod 2) and [Formula: see text] for every [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-critical with respect to [Formula: see text]-odd factor if [Formula: see text] contains a [Formula: see text]-odd factor for every [Formula: see text] with [Formula: see text]. Let [Formula: see text] denote the distance matrix of [Formula: see text]. The largest eigenvalue of [Formula: see text], denoted by [Formula: see text], is called the distance spectral radius of [Formula: see text]. In this paper, we prove an upper bound for [Formula: see text] in a [Formula: see text]-connected graph [Formula: see text] which guarantees [Formula: see text] to be [Formula: see text]-critical with respect to [Formula: see text]-odd factor.
Let FF(e )denote the set of faulty edges in folded hypercube FQ(n) with n >= 3 and |FFe|<= 3n - 8. We prove that there exists a Hamiltonian cycle in FQ(n) - FFe if the following two conditions are satisfied: (1) every vertex in FQ(n) - FFe has degree at least 2, and (2) there is at most one vertex with degree 2 in FQ(n) - FFe.
In this paper, the authors study the concept of domination in the negation of Cayley signed graphs over finite abelian groups. They establish sharp bounds for the domination numbers and characterize the groups that attain these bounds. The study reveals that, for certain Cayley sets, the sum of the domination numbers of a Cayley signed graph and its negation exceeds the group's order. The results are illustrated with several examples.
A colored cut in an edge-colored graph G (c) is a bipartition of its vertex set and its size is the number of colors on edges between the two parts. The Maximum Colored Cut problem seeks to determine the maximum number of colors in colored cuts of G (c), denoted by mcc(G (c)). Say that G (c) is rainbow if no two edges of G (c) have the same color. This problem generalizes the classical Max-Cut problem when G (c) is rainbow. Let G (c) be a p-edge-colored complete graph. In this paper, we show that if G (c) contains no rainbow triangles, then mcc(G (c)) = p. The result holds for the edge-colored complete tripartite graphs without rainbow triangles as well. Moreover, we prove that if G (c) contains no rainbow 4-cycles, then mcc(G (c)) >= 2p/3. We also achieve the same bound if Gc contains no rainbow subdivisions of either K-5 (c) or K-3,K-3 (c).
Let G be a nonempty simple graph, with V (G) denoting its vertex set and E(G) its edge set. For any injective vertex labeling f : V (G) -> & Zopf;, we introduce two corresponding edge labelings: for each edge uv is an element of E(G), let f(-)(uv) = |f(u) - f(v)| and f(+)(uv) = f(u) + f(v). We define the difference index D(G) of G as the minimum size of the range of f(-), and the sum index S(G) as the minimum cardinality of the range of f+. In this work, we establish computational formulas for the sum index and difference index of graphs resulting from graph operations.
Let G be the Cartesian product of n >= 2 paths, with at least one path having an even number of vertices. Let k and r be integers satisfying 2 <= r <= n and 1 <= k <= n - r + 1, and let F be a set of k vertices of G. We prove that there exists an r-factor J of G consisting of k components J(1),J(2),& mldr;,J(k), where each component Ji is an r-regular, r-connected, bipancyclic graph containing exactly one vertex of F. Thus, an n-dimensional grid of even order can be partitioned into regular, connected and bipancyclic subnetworks passing through prescribed vertices. This extends related results for tori and, as a corollary, also applies to hypercubes.
The burning number b(G) of a graph G, introduced by Bonato, is the minimum number of steps to burn the graph. Bonato et al. (2014) conjectured that b(G) <= n for any connected graph G of order n. In this paper, we confirmed this conjecture for 3-caterpillars and 4-caterpillars and then bounded the burning number of p-caterpillars.
Interconnection networks are usually modeled as undirected graphs. Two fundamental classes of interconnection networks structures are paths and cycles, which have desirable properties such as simple structures and low degrees. Therefore, the research about embedding paths and cycles into an interconnection network is a crucial topic. Edge bipancyclicity is a significant class of the cycle embedding problem in interconnection networks. A bipartite graph G(V (G),E(G)) is referred to as bipancyclic if, there exist any possible l-cycles (4 <= l <=|V (G)| and l = 2k,k is an element of & Nopf;) in G. A bipartite graph G(V (G),E(G)) is referred to as edge bipancyclic (respectively, vertex bipancyclic) if, each edge e (respectively, vertex v) of G(V (G),E(G)) lies on all possible l-cycles, where 4 <= l <=|V (G)| and l = 2k (k is an element of & Nopf;). The n (n >= 4)-dimensional wheel network, denoted by CWn, is an important class of Cayley graphs in interconnection networks. The authors prove that CWn (n >= 4) is edge bipancyclic.
Widespread application of parallel and distributed systems in high-performance computing or big data processing has prompted an urgent demand for qualitative and quantitative metrics to evaluate the fault tolerability and vulnerability of underlying interconnection networks for multiprocessor systems. The 1-good-neighbor r-component edge-connectivity is defined as the minimum number of edges whose removal disconnects the network into at least r connected components, with each vertex holding at least one good neighbor. This paper elaborates on the 1-good-neighbor r-component edge-connectivity of the augmented cube AQ(n), denoted as lambda(1,r)(AQ(n)). In detail, for specific ranges of n and r, we determine the lambda(1,r)(AQn), which provides a refined quantitative basis for analyzing the robustness of multiprocessor systems under link failure scenarios.
For a graph G = (V,E), a function f : V ->{0, 1, 2} is a Roman {2}-dominating function (R{2}DF) if f satisfies the following condition: if f(v) = 0, then there is v1, v2 is an element of N(v) such that f(v1) = f(v2) = 1 or there is u is an element of N(v) such that f(u) = 2. Let V if denote the set of vertices assigned i by function f. The weight of a R{2}DF f is the sum f(V ) =& sum;(v is an element of V )f(v). The Roman {2}-domination number is the minimum weight of a R{2}DF on G, denoted by gamma({R2})(G). A function f : V ->{0, 1, 2} is an independent Roman {2}-dominating function (IR{2}DF) if f is a R{2}DF and V-1(f) boolean OR V (f)(2) is an independent set. In this paper, we initially study the influence of edge addition on the Roman {2}-domination number, and for two positive integer a and b, we construct a graph G and an induced subgraph H of G such that gamma{R2}(G) = a and gamma{R2}(H) = b. Based on this construction, we conclude that there is no relation between the Roman {2}-domination number of a graph and its induced subgraph. Furthermore, we give a bound for the independent Roman {2}-domination number on proper interval graphs, and a linear-time algorithm for computing the independent Roman {2}-domination number of unicyclic graphs with restrictions.
As sixth-generation (6G) wireless networks advance, real-time adaptive signal processing is critical for mobility and multi-objective constraints. Traditional methods fall short in dynamic edge environments. We propose a DRL model with dual-agent actor-critic, convolutional features, and mobility forecasting for tasks like modulation and power allocation. It learns in real-time, reducing end-to-end latency by up to 27% relative to tabular RL baselines and by 20% relative to A3C, improving energy efficiency by 13-16% compared to fixed-weight and mobility-unaware configurations, and maintaining low bit error rates in vehicular and aerial simulations. This enables applications in transportation, drones, and IoT.
In wireless sensor networks (WSNs), energy limitations remain a primary challenge to the reliability and longevity of deployed systems, particularly in remote or inaccessible environments. Traditional approaches often treat signal detection and data routing as separate optimization problems, leading to inefficiencies in resource utilization. This paper presents a unified framework that integrates statistical signal detection theory with optimal routing strategies under strict energy constraints. By formulating the problem as a joint statistical optimization task, the model captures the trade-offs between detection accuracy, routing cost, and energy consumption. The proposed approach uses Bayesian decision theory for signal detection and a convex optimization scheme for routing paths that minimize overall energy expenditure while preserving detection performance. Theoretical analyses demonstrate that the joint framework outperforms decoupled strategies in both detection reliability and network lifetime. Comprehensive simulations are conducted on dynamically varying network topologies to evaluate the model's performance under different node densities, noise levels, and energy budgets. Results demonstrate consistent and substantial improvements in packet delivery ratio, false detection rate, and energy consumption compared to benchmark methods, with energy savings exceeding 35% while maintaining detection probability above 0.93. Furthermore, the algorithm adapts to fluctuating signal-to-noise ratios and network degradation over time, making it robust for real-world deployment scenarios such as disaster response, structural health monitoring, and smart agriculture. This integrated approach offers a promising pathway for future WSN designs that demand both accuracy and sustainability.
The in-out degree polynomial of a directed graph is a binary polynomial that combines the indegree polynomial and the outdegree polynomial. In this paper, we give the in-out degree polynomials and their generating function of the directed Fibonacci cubes. As a sequence, we obtain the indegree polynomials and the outdegree polynomials of the cubes. We also study some properties of the coefficients and roots of these univariate polynomials. Finally, we find a close relationship between the indegree polynomials of the cubes and the Morgan-Voyce polynomials, as well as the Fibonacci polynomials.
In this paper, we first show that for every connected graph G, the (1, 2)-rainbow connection number rc(1,2)(G) is upper bounded by rc(1,2)(G[D]) + 3, where D is a connected two-way dominating set of G. As corollaries, we obtain some upper bounds of (1, 2)-rainbow connection number for some special graph classes, including threshold graphs, chain graphs, circular arc graphs, AT-free graphs and interval graphs. In particular, the upper bounds for threshold graphs and interval graphs are tight. We also investigate the (1, 2)-rainbow connection number of G when its complement graph is disconnected by using two-way dominating sets. Furthermore, we show that for every connected graph G, the rc(1,2)(G) is upper bounded by rc(1,2)(G[D]) + 4, where D is a connected two-way two-step dominating set of G. As a corollary, we give an upper bound 3n/delta+1 + 1 of the (1, 2)-rainbow connection number for every connected graph of order n >= 4 and minimum degree delta.
The hypercube Qn is a well-known interconnection network topology due to its symmetry, regularity, and recursive structure. A longstanding open problem posed by Ruskey and Savage asks whether every matching in Qn, for n >= 2, can be extended to a Hamiltonian cycle. Despite over three decades, this problem remains unresolved. However, this conjecture holds for certain special classes of matchings, such as perfect matchings, linear matchings, and some kinds of matchings. It is important to note that not every matching in a hypercube can be extended to a perfect matching. In this study, we contribute to this problem by proving that any matching in the n-dimensional hypercube Qn with at most 4n - 21 edges can be extended to a Hamiltonian cycle, for n >= 6.
The problem initially raised by Ruskey and Savage: whether every matching in an n-dimensional hypercube Q(n) with n >= 2 can be extended to form a Hamiltonian cycle. Fink addressed Kreweras' conjecture by proving that all perfect matchings in Q(n) are extendable to Hamiltonian cycles, and also demonstrated that this extension property holds for all matchings when n = 2, 3, 4. Later, Wang and Zhao expanded the result by showing that every matching in Q(5) can be extended to the Hamiltonian cycle. In this work, we further contribute to this area by proving that every matching consisting of up to 20 edges in the 6-dimensional hypercube Q(6) can be extended to a Hamiltonian cycle.
Let G be a graph with vertex set V (G), M be a subset of V (G), and e be an edge in E(G), and let P(M, e) be the set of pairs (x, y) such that d(G)(x, y) not equal d(G)-e(x, y) where x is an element of M and y is an element of V (G). For a vertex x, let EM(x) be the set of edges e such that there exists a vertex v in G with (x, v) is an element of P({x}, e). M is called a distance-edge-monitoring set if every edge e of G is monitored by some vertex of M, that is, the set P(M, e) is nonempty. The distance-edge-monitoring number of G, denoted by dem(G), is defined as the smallest size of distance-edge-monitoring sets of G. In this paper, for two graphs H and F, we prove that each edge in F-i and H-j can only be monitored by the vertices incident with it, where (R) is the strong product operation and F-i, H-j subset of H (R) F. Moreover, we determine distance-edge-monitoring number of strong product of two specific graphs, including K-n (R) H and Sn (R) H, where H is an element of {K-m, P-m, C-m, S-m, B-m, K-s,K-t}.
Let G be a simple connected graph with vertex set V(G) ={v(1),v(2),...,v(n)}. The distancebetween two vertices v(i )and v(j), denoted by d(G)(v(i), v(j)), is the length of a shortest pathconnecting them inG. The distance matrix of G, denoted by D(G), is the n & times; n matrix (d(v(i),v(j))). Then the distance matrix of a simple connected graphGis a symmetric matrixwith real eigenvalues. The maximum eigenvalue is called the distance spectral radius of G, written as lambda(1)(G). In this paper, we characterize the unique graph whose complementattains the maximum distance spectral radius among all graphs with given connectivity.We also establish bounds for the distance spectral radius of such graph complements.
Metal organic networks (MONs) have played a fundamental structural role in the power sector in the recent years. The investigation of MON construction and formation from different views give us the various uses and benefits in the constructive areas of different fields. The Formation of MONs due to wide changes in metal and organic molecules is popular among the scientist due to their various properties in the different fields. The MON structure has an ability to store the extensive amount of energy due to its core that consists of hydrogen, carbon and many more. The quantitative structure analysis relation and quantitative structure and property relation (QSAR/QSPR) of MONs show the structural feasibility of this chemical compound. Topological indices (TIs) provide the basic and structural information such as melting points, boiling points, chemically stability, pressure, chemical reaction and numerous other fundamental properties. In this paper, we examine the results of different TIs that are degree based for the two different core structures and expansion of MONs.
Let S be a finite commutative semi-ring with 1 not equal 0. We define the unit graph of S, denoted G(S), as a graph where the vertices correspond to the elements of S. Two distinct vertices x and y are said to be adjacent if and only if x + y is an element of U(S), where U(S) denotes the set of units in the semi-ring. In this paper, we determine the metric dimension of the unit graph associated with semi-rings. We also investigate the dominant metric dimension of the unit graph and identify certain semi-rings where the metric dimension and the dominant metric dimension of their unit graphs are equal.
