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The question as stated is false via considering complete bipartite graphs $K_{n,n}$ whose smallest (and only) positive eigenvalue is $n$.

Based on your notation you perhaps meant to ask for the smallest non-negative eigenvalue. This is also false by considering the disjoint union of edges which has spectrum $\pm 1$ for all eigenvalues.

Say you restricted only to connected graphs. The value is still infinite. For example, take the friendship graph $F_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so this makes the sum infinite.

The value is infinite. For example, take the friendship graph $F_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so by considering each of these $G$ in the sum we see that the value must be infinite.

The question as stated is false via considering complete bipartite graphs $K_{n,n}$ whose smallest (and only) positive eigenvalue is $n$.

Based on your notation you perhaps meant to ask for the smallest non-negative value. This is also false by considering the disjoint union of edges which has spectrum $\pm 1$ for all values.

Say you restricted only to connected graphs. The value is still infinite. For example, take the friendship graph $F_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so this makes the sum infinite.

The question as stated is false via considering complete bipartite graphs $K_{n,n}$ whose smallest (and only) positive eigenvalue is $n$.

Based on your notation you perhaps meant to ask for the smallest non-negative eigenvalue. This is also false by considering the disjoint union of edges which has spectrum $\pm 1$ for all eigenvalues.

Say you restricted only to connected graphs. The value is still infinite. For example, take the friendship graph $F_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so this makes the sum infinite.

The question as stated is false via considering complete bipartite graphs $K_{n,n}$ whose smallest (and only) positive eigenvalue is $n$. However, based on your notation you perhaps meant to ask for the smallest non-negative value, which is less obvious to me.

The question as stated is false via considering complete bipartite graphs $K_{n,n}$ whose smallest (and only) positive eigenvalue is $n$.

Based on your notation you perhaps meant to ask for the smallest non-negative value. This is also false by considering the disjoint union of edges which has spectrum $\pm 1$ for all values.

Say you restricted only to connected graphs. The value is still infinite. For example, take the friendship graph $F_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so this makes the sum infinite.