In search of a complete invariant for graphs

Abstract

Let A be the adjacency matrix of an ordinary (simple) graph G and A′=xI+λA+(J-A-I) where I is the n×n identity matrix and J is the n×n matrix of l’s. Then we call P(x,λ)=Per(A′) the permanent polynomial of G. A frame (2-matching) of a graph G is a spanning subgraph F of G whose components are single points, single lines, paths or cycles. If F has w_i paths P_i, i=1,…,n and y_j cycles C_j we let $w(F) = \mathop \Pi \limits_{i = 1}^n p_i ^{w_i } \mathop \Pi \limits_{j = 3}^n c_j ^{y_j }$ the weight of F and call $F(p,c) = \mathop \sum \limits_{F\varepsilon \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{F} } } w(F)$, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{F} }$ is the family of all frames of G, the frame polynomial of G. We conjecture that either of these is a complete invariant for graphs, show their interrelation and present some evidence why the conjectures are plausible.