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Let ∑ni=0(-1)ilixn-i and ∑ni=0(-1)iqixn-i be the characteristic polynomials of the Laplacian matrix and signless Laplacian matrix of an n-vertex graph, respectively. Let αi = |qi - li|, 0 ≤ i ≤ n. In this paper, we find formulas for some of αi's. In particular, we compute αi's for some fullerene graphs. [ABSTRACT FROM AUTHOR]

Leibniz's famous proposition that God has created the best of all possible worlds holds a significant place in his philosophical system. However, the precise manner in which God determines which world is the best remains somewhat ambiguous. Leibniz suggests that a form of "Divine mathematics" is employed to construct and evaluate possible worlds. In this paper, I uncover the underlying mechanics of Divine mathematics by formally reconstructing it. I argue that Divine mathematics is a one-player combinatorial game, in which God's goal is to find the best combination among many possibilities. Drawing on the combinatorial theory, I provide new solutions to some puzzles of compossibility. [ABSTRACT FROM AUTHOR]

Group Theory has diverse applications in mathematics and has been used to solve real-world problems, such as in cryptography, computer graphics, molecular systems biology, and crystal studies. In this article, we delve into a detailed analysis of subgroups of the free group of rank 2. The primary focus of this study is to develop an intuitive visualization for constructing a free set of generators for a free subgroup, using a graph-theoretic approach based on Stallings Foldings to provide a topological interpretation and solution to the problem. This algorithm leverages the connection between algebraic structures and geometric constructs, allowing for an intuitive understanding of the structure of a free group. An implementation of this algorithm is included in Python, making it immediately useful for computational applications. [ABSTRACT FROM AUTHOR]

Phylogenetic trees are used to model evolution: leaves are labeled to represent contemporary species ("taxa""), and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state form a connected subtree. Kelk and Stamoulis [Adv. Appl. Math., 84 (2017), pp. 34--46] showed how to efficiently count, list, and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterized algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we revisit the quantity g2(T), defined as the number of convex characters on T in which each state appears on at least 2 taxa. We use this to give an algorithm with running time Oφn poly(n)), where 1.6181 is the golden ratio and n is the number of taxa in the input trees for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by g2(T) we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to O(1.5895n poly(n)) and obtain a number of new results in themselves relevant to enumeration of matchings on at most binary trees. [ABSTRACT FROM AUTHOR]

Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques. [ABSTRACT FROM AUTHOR]