An element $ x $ in a semigroup is said to be regular if there exists an element $ y $ in the semigroup such that $ x = xyx $. The element $ y $ is said to be a regular part of $ x $. Define the Cayley regularity graph of a semigroup $ S $ to be a digraph with vertex set $ S $ and arc set containing all ordered pairs $ (x, y) $ such that $ y $ is a regular part of $ x $. In this paper, certain classes of Cayley regularity graphs such as complete digraphs, connected digraphs and equivalence digraphs are investigated. Furthermore, structural properties of the Cayley regularity graphs are theoretically applied to study perfect matchings of other algebraic graphs.

Let G and H be graphs. A mapping f from the vertices of G to the vertices of H is known as a $homomorphism$ from G to H if, for every pair of adjacent vertices x and y in G, the vertices $f\left(x\right)$ and $f\left(y\right)$ are adjacent in H. A rectangular grid graph is the Cartesian product of two path graphs. In this paper, we provide a formula to determine the number of homomorphisms from paths to rectangular grid graphs. This formula gives the solution to the problem concerning the number of walks in the rectangular grid graphs.

Let G be a graph with V = V G . A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class { S : S is an independent set of G } and ∅ is denoted by I G . For a graph H , a function f : V ⟶ I H is called an H − independent coloring of G (or simply called an H − coloring) if f x ∩ f y = ∅ for any adjacent vertices x , y ∈ V and f V is a class of disjoint sets. Let α H , G denote the maximum cardinality of the set{ ∑ x ∈ V f x : f is an H − coloring of G }. In this paper, we obtain basic properties of an H − coloring of G and find α H , G of some families of graphs G and H . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph K n and a graph G and prove that α K n □ G = α K n , G .

Northern Thailand frequently suffers from severe PM2.5 air pollution, especially during the dry season, due to agricultural burning, local emissions, and transboundary haze. Understanding how pollution moves across regions and identifying source–receptor relationships are critical for effective air quality management. This study investigated the spatial and temporal dynamics of PM2.5 in northern Thailand. Specifically, it explored how pollution at one monitoring station influenced concentrations at others and revealed the seasonal structure of PM2.5 transmission using network-based analysis. We developed a Python-based framework to analyze daily PM2.5 data from 2022 to 2023, selecting nine representative stations across eight provinces based on spatial clustering and shape-based criteria. Delaunay triangulation was used to define spatial connections among stations, capturing the region’s irregular geography. Cross-correlation and Granger causality were applied to identify time-lagged relationships between stations for each season. Trophic coherence analysis was used to evaluate the hierarchical structure and seasonal stability of the resulting networks. The analysis revealed seasonal patterns of PM2.5 transmission, with certain stations, particularly in Chiang Mai and Lampang, consistently acting as source nodes. Provinces such as Phayao and Phrae were frequently identified as receptors, especially during the winter and rainy seasons. Trophic coherence varied by season, with the winter network showing the highest coherence, indicating a more hierarchical but less stable structure. The rainy season exhibited the lowest coherence, reflecting greater structural stability. PM2.5 spreads through structured, seasonal pathways in northern Thailand. Network patterns vary significantly across seasons, highlighting the need for adaptive air quality strategies. This framework can help identify influential monitoring stations for early warning and support more targeted, season-specific air quality management strategies in northern Thailand.

Let G and H be graphs. A mapping f from V(G) to V(H) is called a weak homomorphism from G to H if f(x) = f(y) or {f(x), f(y)} ∈ E(H) whenever {x, y} ∈ E(G). In this paper, we provide an algorithm to determine the number of weak homomorphisms of paths.