The security of wireless sensor networks is a significant concern and can be achieved by the application of cryptographic algorithms. The symmetric key encryption techniques are widely used cryptographic mechanisms for the security of sensor networks due to its low computational complexity. A symmetric key encryption technique requires a secret key to be shared between both parties for confidential communication. In a wireless sensor network, it is difficult to know which node is going to be in its communication range at the deployment phase. If prior knowledge of sensor location exists, it is an added advantage and helps in the distribution of secret keys among nodes. Even if with the expected location information, distributing the keys properly among the nodes is a challenging task. A proper algorithm must be used so that it gives the adequate utilization of the distributed keys with a minimal number of keys per sensor node. In this paper, we propose a location-dependent key distribution scheme. We use Delaunay Triangulation for the efficient distribution of keys among sensor nodes. The method gives a high probability of secure communication links among nodes with high resilience to the network.

Software Defined Networking (SDN) is a promising paradigm in the field of network technology. This paradigm suggests the separation between the control plane and the data plane which brings flexibility, efficiency and programmability to network resources. SDN deployment in large scale networks raises many issues which can be overcame using a collaborative multi-controller approaches. Such approaches can resolve problems of routing optimization and network scalability. In large scale networks, such as SD-WAN, routing optimization consists of achieving a trade-off between per-flow QoS, the load balancing in each domain as well as the resource utilization in inter-domain links. Multi-Agent Reinforcement Learning paradigm(MARL) is one of the most popular solutions that can be used to optimize routing strategies in SD-WAN. This paper proposes an efficient approach based on MARL which is able to ensure a load balancing among each network as well as optimized resource utilization of inter-domain links. This approach profits from our previous work, denoted SPFLR, and tries to balance the load of the whole network using Deep Q-Networks (DQN) algorithms. Simulation results show that the proposed solution performs better than parallel solutions such as BGP-based routing and random routing.

For a given graph G = (V, E), its integrity is defined as I(G) = minX⊂V {|X|+m(G−X)}, where m(G−X) denote the order of the largest component of G−X. In [9], authors discuss the minimum integrity of tree with given order and the maximum degree. In this paper, we point that the result in [9] is flawed and by elementary method characterize the structure of the minimum integrity tree and thus correct Theorem 4.1 in [9]. Finally, we give the construction method of this kind of extremal graph.

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v1, v2,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

Let R be a finite commutative ring with identity. The co-maximal graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also, Γ2(R) is the subgraph of Γ(R) induced by non-unit elements and [Formula: see text] where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which [Formula: see text], Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.

Tori are important fundamental interconnection networks for multiprocessor systems. Hamiltonian paths are important in information communication of multiprocessor systems, and Hamiltonian path embedding capability is an important aspect to determine if a network topology is suitable for a real application. In real systems, some links may have better performance. Therefore, when embedding Hamiltonian path into interconnection networks, it is desirable that these Hamiltonian paths would pass through the links with better performance. Given a two two-dimensional torus T (m, n) with m, n ≥ 5 odd, let L be a linear forest with at most two edges in T (m, n) and let u and v be two distinct vertices in T (m, n) such that none of the paths in L has u or v as internal node or both of them as end nodes. In this paper, we construct a hamiltonian path of T (m, n) between u and v passing through L.

In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph G with the integer lattice, seen as a graph, in such a way that the resulting hypothetical graph can be traversed using a known algorithm which is based on face routing. To overcome the problem with hypothetical vertices and edges, we develop an algorithm for visiting any k-th neighborhood of a vertex in a graph straight-line drawn in the plane using O(k log k) memory. The memory needed to complete the traversal of a geometric graph then turns out to depend on the maximum graph distance among pairs of distinct vertices of G whose Euclidean distance is greater than one and less than [Formula: see text].

The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the graph with neither perfect matchings nor almost perfect matchings. The fractional perfect matching preclusion and fractional strong perfect matching preclusion are generalizations of the matching preclusion. In this paper, we obtain fractional matching preclusion number and fractional strong matching preclusion number of restricted hypercube-like graphs, which extend some known results.

As spanning subgraphs of hypercubes, exchanged hypercubes contain less edges but maintain lots of desired properties of hypercubes. This paper considers matching preclusion, a kind of measures of edge-fault tolerance, of exchanged hypercubes EH(s, t). We show that EH(s, t) is maximally matched, that is, for s ≥ t, mp(EH(s, t)) = t + 1 and EH(s, t) is super matched if and only if (s, t) ≠ (1, 1). Comparing with results of matching preclusion for hypercubes, we conclude that exchanged hypercubes maintain the matching preclusion property of hypercubes, except for EH(1, 1).

The diagnosability of a multiprocessor system plays an important role. The bubble-sort star graph BSn has many good properties. In this paper, we study the diagnosis on BSn under the comparison model. Following the concept of the local diagnosability, the strong local diagnosability property is discussed. This property describes the equivalence of the local diagnosability of a node and its degree. We prove that BSn (n ≥ 5) has this property, and it keeps this strong property even if there exist (2n − 5) missing edges in it, and the result is optimal with respect to the number of missing edges.