We prove the following necessary conditions for signed graphs on complete graphs to admit additive graceful labelings, thus considerably narrowing down the search for such labelings. If a signed graph on a complete graph \(K_p, ~p\not = 4\), with \(n\) negative and \(m\) positive edges, admits an additively graceful labeling, then (1) \(p\), \(p-2\) or \(p-4\) must be a perfect square, (2) If \(p\) is a perfect square then, both \(m\) and \(n\) are even, (3) If \(p-4\) is a perfect square then, both \(m\) and \(n\) are odd, (4) If \(p-2\) is a perfect square then, \(m\) and \(n\) have opposite parity and (5) The number of odd and even labeled vertices in \(S\) are \(\frac{p+\sqrt{p-i}}{2}\) and \(\frac{p-\sqrt{p-i}}{2}\) according as \(p-i\) is a perfect square, \(i=0,2\) or \(4\). We also provide an example to show that, if a signed graph \(S\) with no negative edges is additively graceful then, \(S\) as a graph need not be graceful. Interestingly, this example also settles 3 more open problems concerning gracefulness and edge consecutive gracefulness (ecg), thereby demonstrating that ecg or \(m\)-gracefulness is not a reliable measure of gracefulness.

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of (3, 6)-fullerenes with V vertices. We also enumerate (3, 6)-fullerenes with mirror symmetry, with 3-fold rotational symmetry, and with both types of symmetry. The resulting formulas are expressed in terms of the prime factorization of V.

What are the collections of sets \({A}_i\subset\mathbb{Z}\) such that any \(n\in\mathbb{Z}\) has exactly one representation as \(n=a_0+a_1+\dotsb\) with \(a_i\in{A}_i\)? The answer for \(\mathbb{N}_0\) instead of \(\mathbb{Z}\) is given by a theorem of de Bruijn. We describe a family of natural candidate collections for \(\mathbb{Z}\), which we call canonical collections. Translating the problem into the language of dynamical systems, we show that the question of whether the sumset of a canonical collection covers the entire \(\mathbb{Z}\) is difficult: specifically, there is a collection for which this question is equivalent to the Collatz conjecture, and there is a well-behaved family of collections for which this question is equivalent to the universal halting problem for Fractran and is therefore undecidable.

We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on n vertices if and only if n ≥ 5, and prove that their underlying graphs must contain 3-cycles. We conjecture that link-irregular tournaments exist if and only if n ≥ 6, providing explicit constructions for n ≤ 8 and computational verification for n ≤ 100. We derive lower bounds on the minimum degree and outdegree required for link-irregularity, establish that almost all link-irregular digraphs are nonplanar, and prove that any link-irregular orientable graph admits a link-irregular labeling. Additionally, we construct explicit examples of link-irregular digraphs with constant outdegree and regular tournaments.

In food processing, effective optimization of process parameters can improve product quality, reduce production costs, and shorten production cycles. This paper improves the traditional particle swarm optimization algorithm by introducing an adaptive learning factor and an elite particle variation strategy, thereby balancing global search and local convergence while preventing particle aggregation and local optima. Using kimchi as a case study, an optimization model including fermentation temperature and other variables is developed, and the improved algorithm is applied to the multi-objective optimization of processing parameters to achieve optimal product quality. Experimental results are compared with those of the traditional particle swarm algorithm and the response surface method. The findings show that the improved model requires only 43 iterations and reduces final risk loss by 14.26%, outperforming the other two methods, which require 52 and 100 iterations, respectively. Thus, the improved particle swarm optimization model can effectively shorten the optimization cycle and reduce the cost of food-processing parameter optimization.