We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs on the same group. Let G be a finite group and let U⊆G∖{e} be symmetric. Writing A(G;U) for the adjacency matrix of Cay(G;U), we prove that, for symmetric subsets S,T,U⊆G∖{e}, A(G;U)=A(G;S)A(G;T) if and only if U=ST and each u∈U has a unique representation u=st with s∈S and t∈T. Equivalently, (∑s∈Ss)(∑t∈Tt)=∑u∈Uu in the group algebra. We also formulate the problem using group convolution: the unique-product condition is equivalent to S−1S∩TT−1={e}. For cyclic groups, we reformulate factorability via mask polynomials and give a Chinese Remainder Theorem reduction for subsets of product form. Finally, we analyze dihedral groups D2n, give non-exhaustive families of factorable triples, and relate strongly symmetric factorizations of D2n to symmetric factorizations of Z2n via the Pêcher map.
We investigate the dynamical behavior of compositions of affine transformations over finite fields from a linear algebraic perspective. For a prime power q and integers n > m > 0, we consider functions of the form F : F-q(n) -> F-q(n) defined by F(x) = T(x + B-t f (Ax)), where T : F-q(n) -> F-q(n), A : F-q(n) -> F-q(m), and B: F-q(n) -> F-q(n-m) are full-rank linear transformations satisfying q the orthogonality condition AB(t) = 0, and f : 1F(q)(m) -> F-q(n-m) is an arbitrary function. This construction, introduced by Gravel and Panario (2019), extends any function f into a permutation over 1Eqn and has applications in cryptographic schemes such as Feistel networks. Our main contribution is a complete characterization of cycle structures-the decomposition of the permutation into disjoint cycles-in terms of the spectral properties of T. When T is the identity matrix, we prove that the functional graph contains only cycles of length 1 and p (where p = char(F-q)), with the number of fixed points given by q(n-m) . |f(-1)(0)|. For general invertible T and constant functions f (x) = y, we establish that all cycle lengths are multiples of a minimal integer d determined by the column space of T-d-I. Using the Jordan canonical form of T and Mobius inversion techniques, we derive explicit formulas for the number of elements in cycles of each length, showing that the possible cycle lengths have the form lcm(S, d, p(m)) where S is a subset of eigenvalue orders and m is bounded by Jordan block sizes. We extend these results to compositions of multiple linear extensions with different functions f(j), proving that the cycle structure depends on the sum Sigma(l)(j =1) f(j) and establishing formulas for cycle lengths in terms of greatest common divisors with the number of iterations. Our results reveal an intricate connection between the algebraic structure of linear transformations and the combinatorial properties of the resulting dynamical system. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Abstract Urban flooding is becoming more frequent and damaging due to the combined effects of climate change and rapid urbanization, exposing the limitations of conventional risk-based flood-management approaches. Consequently, resilience-based strategies have gained attention for managing uncertainty, system complexity, and recovery capacity in urban flood contexts. Flood-resilience index (FRI) frameworks are widely used to operationalize resilience by integrating social, economic, infrastructural, institutional, and environmental dimensions into composite indicators. However, many existing FRI frameworks rely on static or snapshot-based assessments that inadequately represent feedback mechanisms, temporal evolution, and nonstationary flood regimes. This study presents a systematic literature review examining how geographic information systems (GIS), multicriteria decision-making (MCDM), and system dynamics modeling (SDM) have been applied individually and in combination to support urban flood-resilience assessment. Seventy-six peer-reviewed studies published between 2010 and 2024 were analyzed using qualitative content analysis and frequency-based quantitative synthesis. The review synthesizes dominant methodological practices and identifies recurring limitations related to indicator aggregation, weighting, spatial representation, uncertainty treatment, and temporal dynamics. The findings show that GIS–MCDM approaches enable spatially explicit and structured resilience assessment but are commonly constrained by static assumptions and criterion independence. SDM applications offer strong potential for capturing feedback-driven system behavior and for the long-term evolution of resilience, yet are rarely integrated with GIS-based MCDM FRIs. The paper proposes a formalized synthesis-based framework that integrates GIS, MCDM, and SDM to support spatially explicit and dynamically informed urban flood-resilience assessment. It highlights key research gaps for future development.
Motivated by the classification of one-to-one rational functions of low degree in terms of their equivalence classes, we determine all many-to-one (including one-to-one) rational functions of degree two or three on the projective line explicitly in terms of their coefficients. Furthermore, we study the linear-fractional equivalence classes and the value sets of these rational functions. As an application, we characterize two classes of many-to-one quadranomials using their coefficients. These one-to-one quadranomials unify and generalize many results in the literature. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).
We present several Laplacian-type matrices associated with a loopless digraph D: the out-/in-degree Laplacians Lout,Lin, the incidence Laplacian Linc=BBT, and the symmetrized and skew-symmetrized variants Sout,Kout. We show that Linc(D) coincides with the Laplacian of the underlying undirected multigraph, and we derive spectral and characteristic-polynomial relations under arc reversal and complementation (including a simplification for Eulerian digraphs for Sout). We demonstrate that, for simple digraphs, the spectral radius of Lout is bounded above by the order of the digraph and give a characterization in the equality case. We further obtain explicit formulas for joins and line digraphs, giving a general determinantal identity relating the out-degree Laplacian characteristic polynomials of a regular digraph and its line digraph.
