## Sur un nouvel invariant des graphes et un critère de planarité. (On a new graph invariant and a planarity criterion).(English)Zbl 0742.05061

A new graph invariant $$\mu(G)$$ is presented and shown to be monotonic with respect to edge-removal and an operation which includes homeomorphic reduction as a special case. Then, since $$\mu(K_ 5)=\mu(K_{3,3})=4$$, Kuratowski’s planarity criterion is used to show that $$G$$ is planar if and only if $$\mu(G)\leq 3$$. A graph is said to be $$n$$-critical if $$\mu(G)=n$$ and $$\mu(G_ 1)<n$$ for any proper minor $$G_ 1$$ of $$G$$. Then $$n$$-critical graphs for $$0\leq n\leq 4$$ are enumerated, and it is proved that $$G$$ is outer-planar if and only if $$\mu(G)\leq 2$$. It follows from another result that if $$G$$ can be embedded in an orientable surface of genus $$g$$, then $$\mu(G)\leq 4g+3$$. This result could conceivably be used to bound the genus of $$G$$ from below if some procedure could be found to calculate, or at least bound, $$\mu(G)$$. However, the definition given here of $$\mu(G)$$ is highly non-constructive. Let $$O_ G$$ be the set of symmetric $$n\times n$$ matrices derived from the adjacency matrix of $$G$$ by replacing all the 1s by negative reals and the diagonal elements by arbitrary reals. If $$A\in O_ G$$, its smallest eigenvalue $$\lambda_ 1$$ has multiplicity 1; let $$\lambda_ 2$$ be its second-smallest eigenvaue. Then $$\mu(G)$$ is the largest integer $$n _ 0$$ such that there exists a matrix $$A\in O_ G$$ such that $$\lambda_ 2$$ has multiplicity $$n_ 0$$ and satisfies the Arnold hypothesis [V. I. Arnold, Modes and quasimodes, Funct. Anal. Appl. 6, 94-101 (1972; Zbl 0251.70012)]: $$O_ G$$ and the subvariety of symmetric matrices having some eingevalue $$\lambda_ 2$$ with multiplicity $$n_ 0$$ intersect transversally in $$A$$.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory 05C30 Enumeration in graph theory

Zbl 0251.70012
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### References:

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