Fully Indecomposable and Nearly Decomposable Graphs

LetA be an n-square non-negative matrix. If A contains no s\times t zero submatrix, where s + t = n, then it is called fully indecomposable. Also, a graph G is said to be fully indecomposable if its adjacency matrix is fully indecomposable. In this paper we provide some necessary and sucient conditions for a graph to be fully indecomposable. Among other results we prove that a regular connected graph is fully indecomposable if and only if it is not bipartite.  

on the number of cliques and cycles in graphs

We give a new recursive method to compute the number of cliques and cycles of a graph. This method is related, respectively to the number of disjoint cliques in the complement graph and to the sum of permanent function over all principal minors of the adjacency matrix of the graph. In particular, let $G$ be a graph and let $overline {G}$ be its complement, then given the chromatic polynomial of $overline {G}$, we give a recursive method to compute the number of cliques of $G$. Also given the adjacency matrix $A$ of $G$ we give a recursive method to compute the number of cycles by computing the sum of permanent function of the principal minors of $A$. In both cases we confront to a new computable parameter which is defined as the number of disjoint cliques in $G$

Fully Indecomposable and Nearly Decomposable Graphs

LetA be an n-square non-negative matrix. If A contains no s\times t zero submatrix, where s + t = n, then it is called fully indecomposable. Also, a graph G is said to be fully indecomposable if its adjacency matrix is fully indecomposable. In this paper we provide some necessary and sucient conditions for a graph to be fully indecomposable. Among other results we prove that a regular connected graph is fully indecomposable if and only if it is not bipartite.