A vertex u of a graph G = ( V , E ), ve -dominates every edge incident to u , as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve- dominated by at least one vertex of S . The vertex-edge domination number is the minimum cardinality of a ved-set in G . In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.

The rise of IoT has also brought technological developments and threats. Edge computing has become a vital solution for most networks and demands resource-constrained operations. This work presents a lightweight data encryption scheme based on the Sierpinski Triangle Fractal, Chaotic Logistic map and their FPGA implementation on Intel Cyclone IV E. The size of the fractal geometry emerges from the number of bits in the input. Hence, this algorithm is an input-aware cryptosystem that simultaneously handles all input bits using rotation, random data fitting and diffusion. Further, round keys are supplied through the logistic map, and three incarnations of their FPGA implementations were investigated against randomness. Worst-case inputs have been taken to evaluate the design, and the results are validated through standard metrics. Round keys and cipher data have passed the NIST SP 800-22 test suites, thus evidence of the randomness. The proposed cryptosystem consumes only 944 logic elements (0.8%) and cipher 27/128/256 bits of data concurrently on Cyclone IV E FPGA. Encryption time has been calculated as 400 ns for 50 MHz of operating clock.

Let \(G=(V,E)\) be a simple connected graph. The modified Sombor index denoted by \(mSo(G)\) is defined as $$mSo(G)=\sum_{uv\in E}\frac{1}{\sqrt{d^2_u+d^2_v}},$$ where \(d_v\) denotes the degree of vertex \(v\). In this paper we present extremal values of modified Sombor index over the set of trees and unicyclic graphs.

Internet of Things (IoT) devices need to be developed with security considerations in mind for security reasons. Creating quick and safe cryptography modules is necessary for designing secure IoT devices. These modules include the Pseudo Random Number Generator (PRNG), which can be constructed in various ways. Specific techniques include using chaotic maps; in these situations, the chosen chaotic map must be easily implemented in a digital device utilizing the IEEE-754 floating-point standard. Additionally, the chaotic map must produce number sequences with a uniform statistical distribution. This study illustrates the implementation of a PRNG using a Tinkerbell map on FPGA. It takes only a few milliseconds to construct the pseudo-random sequences. The suggested PRNG has proven successful, as demonstrated by NIST SP 800-22 randomization, Hamming, and Shannon-Entropy tests. The resources used to reinforce the security measures reflect 0.73mW of power dissipation and <1% Hardware Utility, thereby increasing the system's overall efficiency.

Let [Formula: see text] be a simple connected graph. The first Zagreb index denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and edge-vertex domination number for trees and characterized the extremal trees attaining the bound.

Let [Formula: see text] be a simple connected graph. The modified first Zagreb index is denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and total domination number for trees and characterize the extremal trees attaining the bound.

Edge data security became the crucial concern of the network connected framework. It demands for the lightweight solutions where the traditional algorithms have not been suitable for the resource constrained devices. Hence, the development of lightweight crypto-solutions has attained the visibility. This proposed leverages the traditional Lightweight Encryption Algorithm (LEA) on reconfigurable hardware such as Field Programmable Gate Array (FPGA) by addressing its potential pitfalls namely vulnerable to differential cryptanalysis. To overcome this, 5-bit Substitution box (S-box) blended with chaos approach has been adopted on the traditional LEA schema. Which also ensures the lightweightness. The substitution and chaotic diffusion processes improves the strength of the LEA to meet out the statistical requirements which was confirmed by conducting the NIST SP 800 – 22 batteries of test by attaining the pass rate of 99.9

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

Let [Formula: see text] be vertices of a graph [Formula: see text] with degree of the vertices being [Formula: see text] and [Formula: see text] respectively. First, let us define the weight of the edge [Formula: see text] as twice the value of [Formula: see text] in [Formula: see text]. Let us define [Formula: see text], the harmonic index of the graph [Formula: see text], as the sum obtained by adding the weight assigned to every edge of [Formula: see text]. In this paper, for the class of trees, we shall obtain an upper bound for the harmonic index [Formula: see text] in terms of the edge-vertex domination number and the order of [Formula: see text]. Also, we shall ascertain that the equality is true by characterizing the collection of all extremal trees attaining this bound.

A vertex [Formula: see text] of a graph [Formula: see text] is said to vertex-edge dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A subset [Formula: see text] is a vertex-edge dominating set (ve-dominating set) if every edge of [Formula: see text] is vertex-edge dominated by at least one vertex of [Formula: see text]. A vertex-edge dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of [Formula: see text], denoted by [Formula: see text], is called the total vertex-edge domination number of [Formula: see text]. In this paper, we prove that for every nontrivial tree of order [Formula: see text], with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize extremal trees attaining the lower bound.

A vertex [Formula: see text] of a graph is said to [Formula: see text]-[Formula: see text] dominate every edge incident with [Formula: see text], as well as every edge incident to vertices adjacent to [Formula: see text]. A subset [Formula: see text] is a total outer connected vertex-edge dominating set of a graph [Formula: see text] if every edge in [Formula: see text] is [Formula: see text]-[Formula: see text] dominated by a vertex in [Formula: see text], the subgraph induced by [Formula: see text] has no isolated vertices and the subgraph induced by [Formula: see text] is connected. We initiate the study of total outer connected vertex-edge domination in graphs. We show that the decision problem for total outer-connected vertex-edge domination problem is [Formula: see text]-Complete even for bipartite graphs. We prove that for every tree of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text] and characterize the trees attaining the lower bound. We also study the effect of edge removal, edge addition and edge subdivision on total outer connected vertex-edge domination number of a graph.

Let [Formula: see text] be a simple graph. A set [Formula: see text] is called a super dominating set if for every vertex [Formula: see text], there exist [Formula: see text] such that [Formula: see text]. The minimum cardinality of a super dominating set of [Formula: see text], denoted by [Formula: see text], is called the super domination number of graph [Formula: see text]. Characterization of trees with [Formula: see text] is presented.

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.

An edge-vertex Roman dominating function (or just ev-RDF) of a graph [Formula: see text] is a function [Formula: see text] such that for each vertex [Formula: see text] either [Formula: see text] where [Formula: see text] is incident with [Formula: see text] or there exists an edge [Formula: see text] adjacent to [Formula: see text] such that [Formula: see text]. The weight of a ev-RDF is the sum of its function values over all edges. The edge-vertex Roman domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of an ev-RDF [Formula: see text]. We provide a characterization of all trees with [Formula: see text], where [Formula: see text] is the domination number of [Formula: see text]

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is the total edge dominating set if every edge in [Formula: see text] is adjacent to at least one edge in [Formula: see text]. The minimum cardinality of a total edge dominated set, denoted by [Formula: see text], is called the total edge domination number of a graph [Formula: see text]. We prove that for every tree [Formula: see text] of diameter at least two with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize the trees attaining each of the bounds.

A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−ℓ+1)/2 ≤ γve(T) ≤(γt(T)+ℓ−1)/2 and characterize trees attaining each of these bounds.

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V, E) is a function f : V (G) → {0, 1, 2} such that for each edge e = uv either max{f (u), f (v)} ≠ 0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f (w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. We characterize trees with vertexedge roman domination number equal to twice domination number minus one.

A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by . We denote the cartesian product of G and H by . In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph , complete graph with complete bipartite graph and wheel graph with complete graph .