/library/oar/handle/123456789/5886 Tue, 23 Dec 2025 12:30:45 GMT 2025-12-23T12:30:45Z /library/oar/handle/123456789/137598 Title: But what is 𝘪, exactly? Demystifying complex numbers, exponential functions and more Authors: Farrugia, Alexander Abstract: But What is i, Exactly? tackles the questions that are not dealt with in the high school mathematics curriculum in a simple and approachable, yet detailed and precise manner. Such questions include: What are complex numbers? Why did people, at some point in history, decide to provide solutions for quadratics which previously did not have any? And what is i, exactly? What is the exponential function? Why is it the number e raised to the power of x? Why e, and not some other number? What is e to the power of ix? How does pi emerge from this function? How do we know that pi is an irrational number? Why do we use radians? What does x to the power of y mean when y is an irrational number, like the square root of 2? What about when it is a complex number? And what is 0 to the power of 0? Why is the cube root of -1 equal to -1 on some calculators, yet on more sophisticated calculators, it is equal to some weird complex number? This book is ideal for students who are currently pursuing, or intend to pursue, a mathematics undergraduate course, giving them a sound understanding of what they have learnt in their high school mathematics. It is also a great reference book for mathematics educators teaching at high school or early undergraduate level to supplement their teaching with material that their curriculum overlooks. Mon, 01 Jan 2024 00:00:00 GMT /library/oar/handle/123456789/137598 2024-01-01T00:00:00Z /library/oar/handle/123456789/137226 Title: The polynomial reconstruction problem for graphs having cut-vertices of degree two Authors: Farrugia, Alexander Abstract: In the polynomial reconstruction problem (PRP), the characteristic polynomial φ(G, x) of a graph G is sought from the polynomial deck PD(G) containing the characteristic polynomials of the n vertex-deleted subgraphs of G. The diagonal entries of the adjugate matrix adj(G, x) of G are the elements of PD(G). The PRP is not completely solved for graphs having vertices of degree one. In this paper, we use adj(G, x) to successfully obtain φ(G, x) from PD(G) for certain graphs having a vertex of degree one whose characteristic polynomial is not discoverable using results from current literature. Our methods require such graphs to have a vertex of degree one adjacent to a vertex of degree two, and this latter vertex would then be a cut-vertex of G. We thus extend this idea to partially solve the PRP for more general graphs that have a cut-vertex of degree two which is not necessarily adjacent to vertices of degree one, presenting an algorithm that provides φ(G, x) from PD(G) when G has such a cut-vertex. Wed, 01 Jan 2025 00:00:00 GMT /library/oar/handle/123456789/137226 2025-01-01T00:00:00Z /library/oar/handle/123456789/137225 Title: Recovering the characteristic polynomial of a graph from entries of the adjugate matrix Authors: Farrugia, Alexander Abstract: The adjugate matrix of G, denoted by adj(G), is the adjugate of the matrix xI − A, where A is the adjacency matrix of G. The polynomial reconstruction problem (PRP) asks if the characteristic polynomial of a graph G can always be recovered from the multiset PD(G) containing the n characteristic polynomials of the vertex-deleted subgraphs of G. Noting that the n diagonal entries of adj(G) are precisely the elements of PD(G), we investigate variants of the PRP in which multisets containing entries from adj(G) successfully reconstruct the characteristic polynomial of G. Furthermore, we interpret the entries off the diagonal of adj(G) in terms of characteristic polynomials of graphs, allowing us to solve versions of the PRP that utilize alternative multisets to PD(G) containing polynomials related to characteristic polynomials of graphs, rather than entries from adj(G). Sat, 01 Jan 2022 00:00:00 GMT /library/oar/handle/123456789/137225 2022-01-01T00:00:00Z /library/oar/handle/123456789/102319 Title: The rank of pseudo walk matrices : controllable and recalcitrant pairs Authors: Farrugia, Alexander Abstract: A pseudo walk matrixWv of a graph G having adjacency matrix A is an nXn matrix with columns (Formula presented.) whose Gram matrix has constant skew diagonals, each containing walk enumerations in G. We consider the factorization over Q of the minimal polynomial m(G, x) of A. We prove that the rank of Wv, for any walk vector v, is equal to the sum of the degrees of some, or all, of the polynomial factors of m(G, x). For some adjacency matrix A and a walk vector v, the pair (A, v) is controllable if Wv has full rank. We show that for graphs having an irreducible characteristic polynomial over Q, the pair (A, v) is controllable for any walk vector v. We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors v, the degree of the minimal polynomial of the largest eigenvalue of A is a lower bound for the rank of Wv. If the rank of Wv attains this lower bound, then (A, v) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that (A, v) is neither controllable nor recalcitrant for any walk vector v. Wed, 01 Jan 2020 00:00:00 GMT /library/oar/handle/123456789/102319 2020-01-01T00:00:00Z