Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages.

Establishing reliable and efficient multicast services in heterogeneous networks under an erasure channel is one of the current priorities in coding theory, particularly in Network Coding (NC) over data networks. Additionally, the increasing number of clients with mobile devices with high processing capabilities and the prevalence of non-delay tolerant traffic have led to the demand for non-feedback multicast schemes regarding distributed resource management. Current communication platforms lack gradual and dynamic coding control based on the type of data being transmitted. This paper proposes a reliable transmission scheme based on a concatenation of systematic coding and Random Linear Network Coding (RLNC) called Fulcrum coding. This hybrid rateless coding scheme with distributed characteristics allows the implementation of an adaptive resource management system to increase the decoding probability during data reception. The method ultimately results in a higher network throughput and shorter transmission times (RTT) per packet by implementing efficient Forward Error Correction (FEC).