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LetGbe ak-connected irregular graph of ordern, sizemand maximum degree. Letbe the signless Laplacian spectral radius ofG. In this article, we prove the following lower bound on: Moreover, we determine similar bounds for the signless Laplacian spectral radius of proper spanning subgraphs andk-edge-connected graphs. [ABSTRACT FROM PUBLISHER]

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined. [ABSTRACT FROM PUBLISHER]

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong. [ABSTRACT FROM AUTHOR]