Enumerative study of RNA secondary structures is one of the most important topics in computational biology. However, most of the existing results are concerned with a single type of structural motifs and are asymptotic. Hairpins and stacks are among the most important motifs in secondary structures. Certain subsets of secondary structures characterized by the number of contained hairpins and the way how these hairpin loops are organized, for instance, cloverleaves (Waterman 1979), have been enumerated in a variety of works, mostly asymptotically. In this paper, we generalize these enumerations and combinatorially obtain exact formulae counting general RNA secondary structures by the joint distribution of hairpins and stacks.

Graph Theory was started by Euler after solving the famous Konigsberg bridge problem. The Graph Coloring is among one of the famous topic for research since it has many beautiful theorems on optimization and its applications in numerous fields of science. The Pi coloring is the coloring of graph parts without a recurring pattern. As a result, it is defined as a function from a set of graph elements with similar properties to the power set of colors, so that each set receives a different color set from the power set. In consequence, Incident Vertex Pi coloring of a graph is defined as the coloring of incident vertices for every single edge with Pi coloring. Incident Vertex Pi coloring of the complete graph is \(n\), wheel graph, star graph and double star graph is \(n+1\), diamond, friendship graphs is \(\Delta +1\), and double fan graph is \(\Delta +2\). In this research, we derived the Incident Vertex PI coloring of Star and Double Star graph’s Middle graph, Total graph, Line graph, and Splitting graph.

For a graph \(G\) with a (not necessarily proper) vertex coloring, a set \(D\subseteq V(G)\) is a polychromatic dominating set of \(G\) if it is a dominating set and each vertex in \(D\) is a different color. The polychromatic domination number of \(G\), \(\rho(G)\), is the minimum number of colors such that, for any \(\rho(G)\)-coloring (with exactly \(\rho(G)\) colors) of the vertices of \(G\), there exists a polychromatic dominating set of \(G\). This paper begins the exploration of the polychromatic domination number. In particular we give tight upper and lower bounds for \(\rho(G)\) both of which are functions of the minimum degree of \(G\).

A novel approach to building strong starters in cyclic groups of orders \(n\) divisible by 3 from starters of smaller orders is presented. A strong starter in \(\mathbb{Z}_n\) (\(n\) odd) is a partition of the set \(\{1,2,\dots,n-1\}\) into pairs \(\{a_i,b_i\}\) such that all pair sums \(a_i+b_i\) are distinct and nonzero modulo \(n\) and all differences \(\pm(a_i-b_i)\) are distinct and nonzero modulo \(n\). A special interest to strong starters of odd orders divisible by 3 is motivated by Horton’s conjecture, which claims that such starters exist (except when \(n=3\) or \(9\)) but remains unproven since 1989. We begin with a starter of order \(p\) coprime with 3 and describe an algorithm to obtain a Sudoku-type problem modulo 3 whose solution, if exists, yields a strong starter of order \(3p\). The process leading from the original to the final starter is called triplication. Besides theoretical aspects of the construction, practicality of this approach is demonstrated. A general-purpose constraint-satisfaction (SAT) solver z3 is used to solve the Sudoku-type problem; various performance statistics are presented.

Let P_n + 1, C_n and S_n represent a path, cycle, and star with n edges, Q_n denote the n-dimensional hypercube graph. The (ℋ₁, ℋ₂)−multidecomposition of G for graphs ℋ₁, ℋ₂, and G is a decomposition of G into copies of ℋ₁ and ℋ₂, where there is at least one copy of ℋ₁ and at least one copy of ℋ₂. In this paper, we prove that the graph Q_n is (S_n − 2, C₄)−multidecomposable for n ≥ 4 and (S_n − 4, P₅)−multidecomposable for n ≥ 5.

Let \(p > 5\) be a prime positive integer, \(m\) and \(s\) be positive integers. We classify the negacyclic codes of length \(5p^s\) over \(R= \mathbb{F}_{p^m}+u\mathbb{F}_{p^m}\), with \(u^2=0\) using the factorisation of cyclotomic polynomials, and we investigate their Hamming distances.

Dominator coloring is a fascinating type of proper coloring where vertices are assigned colors so that every vertex in the graph is within the closed neighborhood of at least one vertex from each color class. The smallest number of colors needed for a dominator coloring is called the dominator chromatic number. In this paper, a new graph product called the closed extended neighborhood corona of two graphs is introduced and its dominator chromatic number for any pair of connected graphs is determined. Also, the dominator chromatic numbers for the extended corona of a path with any graph and a cycle with any graph are derived. Additionally, the dominator chromatic number for the closed neighborhood corona of any two graphs is established.

The article investigates the domination polynomial of generalized friendship graphs. The domination polynomial captures the number of dominating sets of each cardinality in a graph and is known to be NP-complete to compute for general graphs. We establish the log-concave and unimodal properties of these polynomials, and determine their peaks. Furthermore, we analyze the distribution of the zeros of aforesaid polynomial and identify their region in the complex plane. Several open problems are proposed for future exploration.

The generalized Petersen graph \(G(n,k)\) is a cubic graph with vertex set \(V(G(n,k))=\{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}\) and edge set \(E(G(n,k))=\{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}\) where the indices are taken modulo \(n\). Schwenk found the number of Hamiltonian cycles in \(G(n,2)\), and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in \(G(n,3)\) and \(G(n,4)\). This is attained by introducing \(G'(n,k)\), which is a modified version of \(G(n,k)\), and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for \(k=3\) and \(k=4\), which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in \(G(n,k)\) is a sum of the number of admissible subgraphs of \(G'(n,k)\) over a certain subset of the classes.

Perfect codes in the \(n\)-dimensional grid \(\Lambda_n\) of the lattice \(\mathbb{Z}^n\) (\(0<n\in\mathbb{Z}\)) and its quotient toroidal grids were obtained via the truncated distance in \(\mathbb{Z}^n\) given between \(u=(u_1,\cdots,u_n)\) and \(v=(v_1, \ldots,v_n)\) as the graph distance \(h(u,v)\) in \(\Lambda_n\), if \(|u_i-v_i|\le 1\), for all \(i\in\{1, \ldots,n\}\), and as \(n+1\), otherwise. Such codes are extended to superlattice graphs \(\Gamma_n\) obtained by glueing ternary \(n\)-cubes along their codimension 1 ternary subcubes in such a way that each binary \(n\)-subcube is contained in a unique maximal lattice of \(\Gamma_n\). The existence of an infinite number of isolated perfect truncated-metric codes of radius 2 in \(\Gamma_n\) for \(n=2\) is ascertained, leading to conjecture such existence for \(n>2\) with radius \(n\).