Perfect graphs and complex surface singularities with perfect local fundamental group

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

perfect state transfer in unitary cayley graphs over local rings

in this work, using eigenvalues and eigenvectors of unitary cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allowing pst occurring in its unitary cayley graph. moreover, we have some developments when $r$ is a product of local rings.

Perfect Matchings in Edge-Transitive Graphs

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...

Total perfect codes‎, ‎OO-irredundant and total subdivision in graphs

‎Let $G=(V(G),E(G))$ be a graph‎, ‎$gamma_t(G)$. Let $ooir(G)$ be the total domination and OO-irredundance number of $G$‎, ‎respectively‎. ‎A total dominating set $S$ of $G$ is called a $textit{total perfect code}$ if every vertex in $V(G)$ is adjacent to exactly one vertex of $S$‎. ‎In this paper‎, ‎we show that if $G$ has a total perfect code‎, ‎then $gamma_t(G)=ooir(G)$‎. ‎As a consequence, ...

Rank-perfect and" weakly rank-perfect graphs