Subgraphs and the Laplacian spectrum of a graph

AbstractLet G be a graph and H a subgraph of G. In this paper, a set of pairwise independent subgraphs that are all isomorphic copies of H is called an H-matching. Denoting by ν(H,G) the cardinality of a maximum H-matching in G, we investigate some relations between ν(H,G) and the Laplacian spectrum of G

On Hoffman polynomials of λ\lambda-doubly stochastic irreducible matrices and commutative association schemes

Let $\Gamma$ denote a finite (strongly) connected regular (di)graph with adjacency matrix $A$. The {\em Hoffman polynomial} $h(t)$ of $\Gamma=\Gamma(A)$ is the unique polynomial of smallest degree satisfying $h(A)=J$, where $J$ denotes the all-ones matrix. Let $X$ denote a nonempty finite set. A nonnegative matrix B\in{\mbox{Mat}}_X({\mathbb R}) is called {\em $\lambda$-doubly stochastic} if $\sum_{z\in X} (B)_{yz}=\sum_{z\in X} (B)_{zy}=\lambda$ for each $y\in X$. In this paper we first show that there exists a polynomial $h(t)$ such that $h(B)=J$ if and only if $B$ is a $\lambda$-doubly stochastic irreducible matrix. This result allows us to define the Hoffman polynomial of a $\lambda$-doubly stochastic irreducible matrix. Now, let B\in{\mbox{Mat}}_X({\mathbb R}) denote a normal irreducible nonnegative matrix, and ${\cal B}=\{p(B)\mid p\in{\mathbb{C}}[t]\}$ denote the vector space over ${\mathbb{C}}$ of all polynomials in $B$. Let us define a $01$-matrix $\widehat{A}$ in the following way: $(\widehat{A})_{xy}=1$ if and only if $(B)_{xy}>0$ $(x,y\in X)$. Let $\Gamma=\Gamma(\widehat{A})$ denote a (di)graph with adjacency matrix $\widehat{A}$, diameter $D$, and let $A_D$ denote the distance-$D$ matrix of $\Gamma$. We show that ${\cal B}$ is the Bose--Mesner algebra of a commutative $D$-class association scheme if and only if $B$ is a normal $\lambda$-doubly stochastic matrix with $D+1$ distinct eigenvalues and $A_D$ is a polynomial in $B$

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

AbstractFor a nonnegative n×n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A)=n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n×n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials