The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

For a finite group G and a nonempty subset X of G, we construct a graph with a set of vertex X such that any pair of distinct vertices of X are adjacent if they are commuting elements in G. This graph is known as the commuting graph of G on X, denoted by ΓG [X]. The degree exponent sum (DES) matrix of a graph is a square matrix whose (p,q)-th entry is is dvp dvq + dvqdvp whenever p is different from q, otherwise, it is zero, where dvp (or dvq ) is the degree of the vertex vp (or vertex, vq) of a graph. This study presents results for the DES energy of commuting graph for dihedral groups of order 2n, using the absolute eigenvalues of its DES matrix.

Let N = pq be an RSA modulus where p and q are primes not necessarily of the same bit size. Previous cryptanalysis results on the difficulty of factoring the public modulus N = pq deployed on variants of RSA cryptosystem are revisited. Each of these variants share a common key relation utilizing the modified Euler quotient (p 2 − 1)(q 2 − 1), given by the key relation ed − k(p 2 − 1)(q 2 − 1) = 1 where e and d are the public and private keys respectively. By conducting continuous midpoint subdivision analysis upon an interval containing (p 2 − 1)(q 2 − 1) together with continued fractions on the key relation, we increase the security bound for d exponentially. INDEX TERMS Algebraic cryptanalysis, continued fractions method, integer factorization problem, RSA-variants.

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted by 𝑑𝑑𝑣𝑣𝑝𝑝 , is the number of vertices adjacent to vp. The maximum (or minimum) degree matrix of ΓG is a square matrix whose (p,q)-th entry is max{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 } (or min{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 }) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of ΓG for dihedral groups of order 2n, D2n by using the absolute eigenvalues of the corresponding maximum degree matrices (MaxD(ΓG)) and minimum degree matrices (MinD(ΓG)). Here, the comparison of maximum and minimum degree energy of ΓG for D2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.

For a finite group G, let Z(G) be the centre of G. Then the non-commuting graph on G, denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined by an edge whenever vpvq ≠ vqvp. The degree sum matrix of a graph is a square matrix whose (p,q)-th entry is dvp + dvq whenever p is different from q, otherwise, it is zero, where dvi is the degree of the vertex vi. This study presents the general formula for the degree sum energy, EDS (ΓG), for the non-commuting graph of dihedral groups of order 2n, D2n, for all n ≥ 3.