Let \( G = (V, E) \) be a simple graph with vertex set \( V \) and edge set \( E \). If \( k \geq 2 \) is an integer, then the signed edge \( k \)-independence function of \( G \) is a function \( f : E \to \{-1, 1\} \) such that \(\sum_{e’ \in N[e]} f(e’) \leq k – 1\) for each \( e \in E \). The weight of a signed edge \( k \)-independence function \( f \) is \(\omega(f) = \sum_{e \in E} f(e).\) The signed edge \( k \)-independence number \( \alpha_k^s(G) \) of \( G \) is the maximum weight of a signed edge \( k \)-independence function of \( G \). In this paper, we initiate the study of the signed edge \( k \)-independence number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.

Let \( S = S_1 S_2 S_3 \dots S_n \) be a finite string which can be written in the form \( X_1^{k_1} X_2^{k_2} \dots X_r^{k_r} \), where \( X_i^{k_i} \) is the \( k_i \) copies of a non-empty string \( X_i \) and each \( k_i \) is a non-negative integer. Then, the curling number of the string \( S \), denoted by \( \text{cn}(S) \), is defined to be \( \text{cn}(S) = \max\{k_i : 1 \leq i \leq r\} \). Analogous to this concept, the degree sequence of the graph \( G \) can be written as a string \( X_1^{k_1} \circ X_2^{k_2} \circ X_3^{k_3} \circ \dots \circ X_r^{k_r} \). The compound curling number of \( G \), denoted \( \text{cn}^c(G) \), is defined to be \(\text{cn}^c(G) = \prod_{i=1}^{r} k_i.\) In this paper, the curling number and compound curling number of the powers of the Mycielskian of certain graphs are discussed.

The symmetric inverse monoid, \(\text{SIM}(n)\), is the set of all partial one-to-one mappings from the set \(\{1, 2, \dots, n\}\) to itself under the operation of composition. Earlier research on the symmetric inverse monoid delineated the process for determining whether an element of \(\text{SIM}(n)\) has a \(k\)th root. The problem of enumerating \(k\)th roots of a given element of \(\text{SIM}(n)\) has since been posed, which is solved in this work. In order to find the number of \(k\)th roots of an element, all that is needed is to know the cycle and path structure of the element. Conveniently, the cycle and cycle-free components may be considered separately in calculating the number of \(k\)th roots. Since the enumeration problem has been completed for the symmetric group, this paper only focuses on the cycle-free elements of \(\text{SIM}(n)\). The formulae derived for cycle-free elements of \(\text{SIM}(n)\) here utilize integer partitions, similar to their use in the expressions given for the number of \(k\)th roots of permutations.

Motivated by finding a way to connect the Roman domination number and 2-domination number, which are in general not comparable, we consider a parameter called the Italian domination number (also known as the Roman \((2)\)-domination number). This parameter is bounded above by each of the other two. Bounds on the Italian domination number in terms of the order of the graph are shown. The value of the Italian domination number is studied for several classes of graphs. We also compare the Italian domination number with the 2-domination number.

The 3-sphere regular cellulation conjecture claims that every 2-connected cyclic graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere. The conjecture is obviously true for planar graphs. 2-connectivity is a necessary condition for a graph to satisfy such a property. Therefore, the question whether a graph is the 1-dimensional skeleton of a regular cellulation of the 3-dimensional sphere would be equivalent to the 2-connectivity test if the conjecture were proved to be true. On the contrary, it is not even clear whether such a decision problem is computationally tractable.

We introduced a new class of graphs called weakly-split and proved the conjecture for such a class. Hamiltonian, split, complete \( k \)-partite, and matrogenic cyclic graphs are weakly split. In this paper, we introduce another class of graphs for which the conjecture is true. Such a class is a superclass of planar graphs and weakly-split graphs.

The maximum number of internal disjoint paths between any two distinct nodes of faulty enhanced hypercube \( Q_{n,k} (1 \leq k \leq n-1) \) are considered in a more flexible approach. Using the structural properties of \( Q_{n,k} (1 \leq k \leq n-1) \), \( \min(d_{Q_{n,k}-V}(x), d_{Q_{n,k}-V}(y)) \) disjoint paths connecting two distinct vertices \( x \) and \( y \) in an \( n \)-dimensional faulty enhanced hypercube \( Q_{n,k}-V (n \geq 8, k \neq n-2, n-1) \) are conformed when \( |V’| \) is at most \( n-1 \). Meanwhile, it is proved that there exists \( \min(d_{Q_{n,k}-V}(x), d_{Q_{n,k}-V}(y)) \) internal disjoint paths between \( x \) and \( y \) in \( Q_{n,k}-V (n \geq 8, k \neq n-2, n-1) \), under the constraints that (1) The number of faulty vertices is no more than \( 2n-3 \); (2) Every vertex in \( Q_{n,k}-V’ \) is incident to at least two fault-free vertices. This results generalize the results of the faulted hypercube \( FQ_n \), which is a special case of \( Q_{n,k} \), and have improved the theoretical evidence of the fact that \( Q_{n,k} \) has excellent node-fault-tolerance when used as a topology of large-scale computer networks, thus remarkably improving the performance of the interconnect networks.

Given a finite non-empty sequence \( S \) of integers, write it as \( XY^k \), consisting of a prefix \( X \) (which may be empty), followed by \( k \) copies of a non-empty string \( Y \). Then, the greatest such integer \( k \) is called the curling number of \( S \) and is denoted by \( cn(S) \). The notion of curling number of graphs has been introduced in terms of their degree sequences, analogous to the curling number of integer sequences. In this paper, we study the curling number of certain graph classes and graphs associated to given graph classes.

In this paper, we investigate and obtain the properties of higher-order Daehee sequences by using generating functions. In particular, by means of the method of coefficients and generating functions, we establish some identities involving higher-order Daehee numbers, generalized Cauchy numbers, Lah numbers, Stirling numbers of the first kind, unsigned Stirling numbers of the first kind, generalized harmonic polynomials and the numbers \( P(r, n, k) \).

In this paper, we give the sufficient conditions for a graph with large minimum degree to be \( s \)-connected, \( s \)-edge-connected, \( \beta \)-deficient, \( s \)-path-coverable, \( s \)-Hamiltonian and \( s \)-edge-Hamiltonian in terms of spectral radius of its complement.

The maximum number of clues in an \( n \times n \) American-style crossword puzzle grid is explored. Grid constructions provided for all \( n \) are proved to be maximal for all even \( n \). By using mixed integer linear programming, they are verified to be maximal for all odd \( n \leq 49 \). Further, for all \( n \leq 30 \), all maximal grids are provided.