In this paper some concepts used in graph theory are introduced, such as directed, undirected, connected, tree, regular or gradient, divergent or Laplacian graphs, and relationships between the diameter of the graph, or the largest second proper value of its adjacency matrix, with respect to the Cheeger constant to identify expander graphs k-regular. With these guidelines defined, some properties are introduced in Cayley graphs, with illustrative examples, and methodologies to identify if the corresponding graph is k-regular or a directed tree. Finally, Cayley expander graphs are related to their diameter or the second largest eigenvalue.

In this paper some concepts used in graph theory are introduced, such as directed, undirected, connected, tree, regular or gradient, divergent or Laplacian graphs, and relationships between the diameter of the graph, or the largest second proper value of its adjacency matrix, with respect to the Cheeger constant to identify expander graphs k-regular. With these guidelines defined, some properties are introduced in Cayley graphs, with illustrative examples, and methodologies to identify if the corresponding graph is k-regular or a directed tree. Finally, Cayley expander graphs are related to their diameter or the second largest eigenvalue.

A remarkable feature of some insects is the synchronized emergence of a large number of insects, such as cicadas that appear in bud ratios every 13 to 17 years. Therefore, this paper presents several mathematical models that explain such synchronized evolutions for certain life cycles in the cicada. For this purpose, a discrete model with lags, proposed by Hoppensteadt and Keller, is taken as a reference, where the population for a type of cicada incorporates a predator species and a carrying capacity in the environment. A qualitative analysis of this system is carried out to determine the existence of synchronized solutions, and the results are contrasted with other models.