The normalized Laplacian and related indexes of graphs with edges blew up by cliques

Vector clique decompositions

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set. A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible. Let ${\bf v} \in {\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $ν_{\bf v}(L) = \sum_{F \in F_k} {\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic to $…

环多项式：从Tutte的普适V-函数到双区带几何

The Loopy Polynomial: from Tutte's Universal $V$-Function to Bizonotopal Geometry

研究环多项式L_G，证明其包含Tutte多项式并统一多个图不变量，提出与U多项式区分能力等价的猜想，并解决Merino和Noble的开放问题。