For a graph X without isolated vertices and without isolated edges, a product-irregular labelling ω : E(X) → {1, 2,. .. , s}, first defined by Anholcer in 2009, is a labelling of the edges of X such that for any two distinct vertices u and v of X the product of labels of the edges incident with u is different from the product of labels of the edges incident with v. The minimal s for which there exists a product irregular labeling is called the product irregularity strength of X and is denoted by ps(X). Clique cover number of a graph is the minimum number of cliques that partition its vertex-set. In this paper we prove that connected graphs with clique cover number 2 or 3 have the product-irregularity strength equal to 3, with some small exceptions.

For a graph X without isolated vertices and without isolated edges, a product-irregular labelling ω : E(X) → {1, 2, . . . , s}, first defined by Anholcer in 2009, is a labelling of the edges of X such that for any two distinct vertices u and v of X the product of labels of the edges incident with u is different from the product of labels of the edges incident with v. The minimal s for which there exist a product irregular labeling is called the product irregularity strength of X and is denoted by ps(X). Clique cover number of a graph is the minimum number of cliques that partition its vertex-set. In this paper we prove that connected graphs with clique cover number 2 or 3 have the product-irregularity strength equal to 3, with some small exceptions.

In this paper, we present a plugin for visualising vote based consensus mechanisms primarily aimed to help engineers understand and debug blockchain and distributed ledger protocols. Both tools are built as Grafana plugins and make no assumptions on the data storage implementation. The plugins can be configured via Grafana plugin configuration interface to fit the specifics of the protocol implementation.