Let Delta(n )denotes the triangular ladder graph. The multiplicative degree Kirchhoff index of a graph G with m number of edges is Kf(& lowast;)(G) = 2m & sum;( n)(i-2 )1/lambda (i). The term lambda(i )are the normalized Laplacian (NL) spectrum. In this study we have calculated the Kf(& lowast;)(G) through the lambda(i) spectrums for Delta(n) network. To compute our main results, we have used the techniques of decomposition theorem (DT).
Reliability analysis is important for the design of large multiprocessor systems. The connectivity of a graph is an important parameter for assessing the reliability of interconnection networks. For a connected graph G and F subset of V (G), if G - F is disconnected and there exist at least r components and each vertex v is an element of V (G - F) has at least g neighbors, then F is called a g-good r-component cut of G. The g-good r-component connectivity of G, denoted by c kappa(g,r)(G), is the minimum cardinality of g-good r-component cuts of G. In this paper, we determine the 2-good 3-component connectivity of n-dimensional alternating group networks ANn, that is c kappa(2,3)(AN(n)) = 6n - 19 for n >= 5.
In network design, transmission delay and fault tolerance are two important parameters. They are typically measured by diameter and connectivity, respectively. In order to measure them more accurately, the fault diameter and wide diameter were proposed. Locally exchanged twisted cube LeTQ(s,t) is a network which has the advantages of the locally twisted cube and the exchanged hypercube. It has many excellent properties, such as small diameter and low overhead. In this work, we obtain the lower and upper bounds of the (s + 1)-wide diameter and s-fault diameter of LeTQ(s,t), which are inverted right perpendiculars+3/2inverted left perpendicular + inverted right perpendiculart+3/2inverted left perpendicular + 3 and inverted right perpendiculars+3/2inverted left perpendicular + inverted right perpendiculart+3/2inverted left perpendicular + 5 for 3 <= s <= t, respectively.
In this paper, we introduce the concept of the geodetic coalition graph derived from a given graph G and its geodetic coalition partition. The geodetic coalition graph captures the interactions between geodetic coalition partners. We investigate structural properties of geodetic coalition graphs and compute them for standard graph families including paths, cycles, complete graphs and trees. A real-world application is demonstrated by modeling delivery hub systems using geodetic coalition graphs, showcasing their potential in optimizing route coverage and connectivity. This study extends the utility of geodetic principles in coalition based graph modeling.
The notion of the Parikh matrix was introduced by Mateescu (2001) to study the numerical properties of a word over an alphabet in terms of subwords. The Parikh determinant for a word has recently been introduced to study the analogy of the classical notion of determinant in matrix theory. The Parikh determinant of a word w over an alphabet Sigma = {a(1) < a(2) < & ctdot; < ak} can be computed by simply finding the number of occurrences of the scattered subword a(k)a(k-1)& mldr;a(1), i.e., |w|a(k)a(k-1)& mldr;a(1). This paper investigates certain Parikh determinant properties in relation to word operations such as partial sum, product, circular variance of words and SShuffle operators etc. A necessary and sufficient condition, for the Parikh determinants of two conjugate ternary words to be equal is also provided.
The restricted connectivity is a conditional connectivity as an improvement of the classical connectivity to measure the fault-tolerance of interconnection networks. For a graph G = (V (G),E(G)), its h-restricted connectivity kappa(h)(G) =min{|S||S subset of V (G)} such that G - S is disconnected and each vertex in G - S with degree at least h. The n-dimensional bubble-sort star graph BSn is an interconnection network, which possesses many favourable properties including regularity, recursive construction, vertex symmetry, and high fault tolerance. In this paper, we show that kappa(3)(BSn) = 12n - 36 for n >= 5, which is an improvement of kappa(2)(BSn) = 8n - 22 for n >= 5 [Discrete Appl. Math. 217 (2017) 691-706.].
The concept of “relative difference family”, denoted by [Formula: see text]-RDF, was introduced by M. Buratti in 1998. Especially cyclic RDFs with [Formula: see text] are closely related to the concept of “optical orthogonal codes”. However, to the best of our knowledge, few studies on partitioned RDFs with [Formula: see text] have been reported in the literature to date. In this paper, we establish two results to characterize partitioned RDFs, among which a lower bound is used to achieve optimality in applications and determine the size of base blocks of partitioned RDFs. As the main contribution, two classes of partitioned uniform cyclic RDFs with [Formula: see text] are proposed by means of some specific mappings. Employing these partitioned uniform cyclic RDFs, we obtain regular and optimal difference systems of sets with flexible parameters at the same time.
The jump graph J(G) of a graph G of order n >= 3 is the complement graph of the line graph L(G). The line graph L(G) of G is the graphical realisation of edge adjacency in G and the jump graph is the graphical realisation of edge independence in G. In this paper, coloring related harmonic polynomials and topological indices of jump graphs of paths and certain cycle related graphs are discussed.
This paper studies the online problem of search and rescue on a (unit) ring. Assuming that an object is located somewhere unknown on the ring, we aim to find the object and deliver it to a designated destination by one or more mobile agents, so that the total time spent is minimized. The agent(s) can only move along the ring, and is initially located a distance of a is an element of (0, 1/2] away from the destination. For single-agent search and rescue, we present an optimal deterministic algorithm with a competitive ratio of min{ 2-a /1-a , a+ root a(2)+4a /2a }, and as well as randomized algorithms corresponding to a = 1/2 and a is an element of (0, 1/2), whose expected competitive ratios can be accurately computed by non-linear programming. For the two-agent search and rescue, we propose deterministic algorithms with competition ratios depending on whether the agents can communicate through the radios
In this paper, we investigate two classes of complete permutation polynomials over finite fields F-33n. The first class of complete permutation polynomial is of the form H(Tr-n(n)3n(beta x)) + Tr-n(3n)(b(1)x) +alpha Tr-n(3n)(b(2)x) +alpha Tr-2(n)3n(b(3)x). In this class, for a fixed beta is an element of F-33n, We find four values of (b) over bar = (b(1),b(2),b(3)) (b(i) is an element of F-33n) for which H(Tr-n(3n)(beta x)) + Tr-n(3n)(b(1)x) +alpha Tr-n(3n)(b(2)x) +alpha Tr-2(n)3n(b(3)x) is a complete permutation polynomial over F-33n for every polynomial H(x)is an element of F-3n[x]. The second class of complete permutation polynomial is of the form H{Tr-n(3n)(gamma x)}+alpha Tr-n(3n)(c(1)x)+alpha Tr-2(n)3n(c(2)x). We characterize this class for several specific values of c(1) and c(2).
For a connected graph G(V, E), the total outer-connected dominating set problem (TOCD) asks for a partition of V(G) into D and V(G)\D such that D is a total dominating set of G and G[V(G)\D] is connected. TOCD is NP-complete on general graphs, chordal graphs and split graphs [1]. In this paper, we introduce a subclass of split graphs called star-convex split graphs and strengthen the NP-completeness result of split graphs; TOCD is NP-complete on star-convex split graphs with convexity on clique (independent set). In the parameterized setting, it is interesting to note that the total outer-connected domination problem, with respect to the solution size, is W[2]-hard when the convexity of star-convex split graphs lies on the clique, whereas it is W[1]-hard when the convexity lies on the independent set. Further, we obtain an interesting dichotomy for TOCD on star-convex split graphs with convexity on clique; TOCD is polynomial-time solvable if no pendant vertices, and NP-complete, otherwise. If the convexity is on the independent set, then the dichotomy is; on K-1,K-6-free star-convex split graphs, TOCD is NP-complete, and polynomial-time solvable on K-1,K-5-free split graphs. It is natural to ask for the status of approximation algorithm for finding minimum total outer-connected domination on star-convex split graphs with convexity. We prove that star-convex split graphs with convexity on independent set does not admit (1-e) ln V(G)-approximation algorithm unless NP subset of DTIME(n(O (log log n))). Furthermore, we prove that for path-convex split graphs finding a minimum total outer-connected domination is polynomial-time solvable.
We investigate coupled task scheduling problems under constrained resource availability on a single machine, which are of fundamental interest and already NP-hard even in this basic setting. In this model, every job j comprises two tasks. Their processing times are distinct, and they are separated by a fixed time interval. For the second task, the job-specific processing time depends on the amount of resource allocated to it. The aim is to minimize the makespan or the total completion time, under the resource constraints. For the goal of minimizing the makespan, we develop a 7/2-approximation algorithm for the general case. Then we propose a 3-approximation algorithm for the special case where the time intervals are all equal. For the goal of minimizing the total completion time, we propose a 3-approximation algorithm when all time intervals are equal. Finally, we propose a 2-approximation algorithm when the processing time of the first task is equal to the time interval.
The construction of vertex (edge) disjoint paths have been well applied to the study of connectivity, diameter, parallel routing, reliability and fault tolerance of an interconnection network. The Menger-type problem about vertex (edge) disjoint paths in interconnection network has received extensive attention. A connected graph is strong Menger vertex (edge) connected if there exist min{d(G)(x), d(G)(y)} vertex (edge) disjoint paths between any two distinct vertices x and y in graph G. In this work, we determine the strong Menger connectivity of alternating group network (AN(n)) when the genetic subnetwork fails. In detail, we show that AN(n) - H is strong Menger vertex (edge) connected where H is isomorphic to AN(s) (3 <= s <= n - 1).
The server consolidation problem, which aims to allocate services into fewer servers, is one of the main concerns in cloud computing. In the online setting where services are released sequentially, we study this problem by modeling it as an online fault-tolerant bin packing problem: Bins and items respectively represent servers with uniform capacity and services with heterogeneous workloads. To tolerate up to f faulty servers, each service is replicated into f + 1 replicas including one primary and f standby replicas. The target of the considered bin packing problem is to minimize the total number of required bins while ensuring that each service retains at least one available replica despite faults in any f bins.Our main work is to propose a Harmonic-based algorithm and prove that this algorithm achieves an asymptotic competitive ratio less than 1.694. This performance surpasses the previous best ratio of 1.75 and closely approaches the theoretical lower bound of 1.691 for classic Harmonic algorithm.
In this article, we determine the extremal graphs concerning the augmented Sombor index over the connected graph (resp. chemical graph) of a given order. Then we have determined several upper and lower bounds on the augmented Sombor index of a graph with given parameters. Finally, in Sec. 3, we determine the relationships between the augmented Sombor index and other standard topological indices and also characterize the family of graphs where the bounds are reached.
In the paper, we study the All-or-Nothing Resource Allocation problem with Delay Constraints (AoNRA-DC). We are given a set of servers and a set of clients, where each server has a resource supply and each client has a demand for resource. The objective is to maximize the number of clients whose demands are fully satisfied by servers within the delay constraint. Importantly, if a client's demand is not completely satisfied, the client is considered unserved. Moreover, clients may be served by multiple servers, and servers may serve multiple clients simultaneously. By reducing to the maximum set packing problem, we show that AoNRA-DC is NP-hard and cannot be approximated within any constant factor. Next, observing that in many practical scenarios the delay function satisfies the metric property (i.e., the triangle inequality), we present an approximation algorithm with a bifactor ratio of (13, 5) based on linear programming rounding. That means, our approach guarantees that the solution serves at least one-third as many clients as the optimal solution while ensuring that the delay remains within five times the predefined threshold.
Weapon-Target Assignment (WTA) is a cornerstone of coordinated air combat operations, where efficient allocation strategies are essential for mission success. This paper addresses the dynamic WTA problem by formulating a novel mathematical model that incorporates hard constraints on weapon allocation quantities to prevent resource wastage. To solve this complex optimization challenge, we propose a Priority-constrained Particle Swarm Optimization (PPSO) algorithm. Experimental results demonstrate that the PPSO algorithm not only achieves superior target assignment accuracy but also significantly minimizes weapon consumption. Furthermore, the algorithm exhibits robust adaptability to practical operational constraints, such as limited weapon inventory and fuel capacity, in cooperative air combat scenarios.
The topological structure of almost all interconnection networks is constructed in a recursive manner. For an n-dimensional recursive network G(n) can be constructed by some different copies of G(n-1) by adding some edges among them. The reliability and stability of recursive networks are usually measured by the t-embedded edge connectivity. For an n-dimensional recursive network G(n), the t-embedded edge connectivity of G(n), denoted by eta(t)(G(n)), is the minimum size of edges whose removal disconnects G(n) and each vertex in the resulting network is in a t-dimensional subnetwork G(t). The n-dimensional alternating group network AN(n) is a typical recursive network. In this paper, we show that eta(t)(AN(n)) = (n - t) t!/2 for n >= 4 and 3 <= t <= n - 1.
We propose a novel scheduling problem with connectivity constraints. This problem defines a connected graph G = (V,E), where each edge e is an element of E is associated with a processing time t(e) and cost c(e), i.e., t,c : E -> & Ropf;(+) boolean OR{0}, where & Ropf;(+) is the set of real positive numbers. The objective is to select a path P in G to establish connectivity between two given vertices and further to schedule the corresponding edges on m parallel machines for minimizing the sum of the makespan and the total cost. This problem is strongly NP-hard, combining elements of graph theory and parallel machine scheduling. In this paper, we focus on the uniform parallel machine environment, for which we develop approximation algorithms and provide worst-case analysis. Our results contribute to scheduling theory in combinatorial optimization and provide new insights into network restoration under resource constraints.
With the rapid development of multiprocessor systems, fault tolerability plays a vital role in measuring the reliability of multiprocessor systems. The traditional connectivity and conditional connectivity are excellent indicators to measure the reliability of multiprocessor system's underlying network. However, the cardinality and the number of connected components of the survival network are indispensable to measure the robustness of the interconnection networks. Therefore, g-extra H-structure connectivity has been proposed. Motivated by the g-extra H-structure connectivity, the g-extra H-structure diagnosability is proposed in this paper to evaluate the fault diagnostic capability of multiprocessor systems successively. In this paper, we determine the g-extra H-structure connectivity and diagnosability of hypercube, where H congruent to K1, 1 and g is an element of {0, 1, 2, 3}. Furthermore, we develop a novel fault diagnosis algorithm to detect all faulty units and demonstrate its efficiency through extensive experimental evaluations.
