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Under an edge coloring (not necessarily proper), a rainbow path is a path whose edge colors are all distinct. The d-local rainbow connection number lrcd(G) (respectively, d-local strong rainbow connection number lsrcd(G)) is the smallest number of colors needed to color the edges of G such that any two vertices with distance at most d can be connected by a rainbow path (respectively, rainbow geodesic). This generalizes rainbow connection numbers, which are the special case d = diam(G). We discuss some bounds and exact values. Moreover, we also characterize all triples of positive integers d, a, b such that there is a connected graph G with lrcd(G) = a and lsrcd(G) = b.

The Graph Neural Network (GNN) is an advanced use of graph theory that is used to address complex network problems. The application of Graph Neural Networks allows the development of a network by the modification of weights associated with the vertices or edges of a graph G (V, E). Data encryption is a technique used to improve data security by encoding plain text into complex numerical configurations, hence minimizing the probability of data leaking. This study seeks to explain the potential of improving data security through the application of graph neural networks and transposition techniques for information manipulation. This study involves an algorithm and simulation that discusses the use of the transposition approach in manipulating information. This is accomplished by the implementation of a graph neural network, which develops the interaction between vertices and edges. The main result of this research shows empirical evidence supporting the notion that the length of the secret key and the number of characters utilized in data encryption have a direct impact on the complexity of the encryption process, hence influencing the overall security of the created data.

NTRU is a lattice-based public-key cryptosystem designed by Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman in 1996. NTRU published on Algorithmic Number Theory Symposium (ANTS) in 1998. The ANTS’98 NTRU became the IEEE standard for public key cryptographic techniques based on hard problems over lattices in 2008. NTRU was later redeveloped by NTRU Inc. since 2018 and became one of the finalists in round 3 of the PQC (Post-Quantum Cryptography) standardization process organized by NIST in 2020. There are two types of NTRU algorithms proposed by NTRU Inc., which are classified based on parameter determination, NTRU-HPS (Hoffstein, Pipher, Silverman) and NTRU-HRSS (Hulsing, Rijnveld, Schanck, Schwabe). Algebraic attacks on ANTS’98 NTRU had previously been carried out in 2009 and 2012. In this paper, algebraic attack is performed on the renewed NTRU submitted to the third round of NIST PQC standardization in 2020. This study aims to obtain system of polynomial equations representing plaintext bits of NTRU-HPS and NTRU-HRSS which can be used to find the private key in the next study. As a result, some polynomial equations with unknown variables were obtained.

Let G = (V,E) be a graph with a vertex set V and an edge set E of G, with order n. Modular irregular labeling of a graph G is an edge k-labeling φ:E → {1, 2,…,k} such that the modular weight of all vertices is all different. The modular weight is defined by wtφ(u) = Σv∈N(u) φ(uv) (mod n). The minimum number k such that a graph G has modular irregular labeling with the largest label k is called modular irregularity strength of G. In this research, we determine the modular irregularity strength for a disjoint union of cycle graph, (m C n )= mn 2 +1 $ (mC_{n})=\frac{mn}{2}+1 $ for n ≡ 0 (mod 4), a disjoint union of sun graph, ms(m(Cn ⊙ K1))2 = ∞ for n and m even and ms(m(Cn ⊙ K1)) = mn otherwise, and a disjoint union of middle graph of cycle graph, ms(mM(Cn)) = ∞ for n and m both odd numbers and (mM( C n ))= mn 2 +1 $(mM({C_n})) = {{mn} \over 2} + 1$ otherwise.

A distance antimagic labeling of graph G = (V, E) of order n is a bijection f: V(G) → {1, 2, . . ., n} with the property that any two distinct vertices x and y satisfy ω(x), ω(y), where ω(x) denotes the open neighborhood sum PΣaN(x)f(a) of a vertex x. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph C(n; S) is a Cayley graph with order n and generating set S, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs C(n; {1, k}) with odd n. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a simple, connected, and undirected graph, where V and E are the set of vertices and the set of edges of G. An edge magic total labeling on G is a bijection f: V → E → {1, 2, ..., |V | + |E|}, provided that for every uv ∈ E,w(uv) = f(u) + f(v) + f(uv) = K for a constant number K. Such a labeling is said to be a super edge magic total labeling if f(V) = {1, 2,..., |V |} and a b-edge consecutive edge magic total labeling if f(E) = {b + 1, b + 2, ..., b + |E|} with b ≥ 1. In this research, we give sufficient conditions for a graph G having a super edge magic total labeling to have a b-edge consecutive edge magic total labeling. We also give several classes of connected graphs which have both labelings. [ABSTRACT FROM AUTHOR]