An upper bound on the algebraic connectivity of outerplanar graphs

In this paper, we determine upper bounds on the algebraic connectivity, denoted as a(G), of maximal outerplanar graphs. We show that if G is a maximal outerplanar graph on n≥12 vertices not of the form K1∨Pn−1, then a(G)≤1 with equality holding for exactly two maximal outerplanar graphs on 12 vertices. We show this by assigning labels y1,…,yn to the vertices and showing the existence of vertex labellings such that ∑uv∈E(G)(yu−yv)2∕∑v∈V(G)yv2<1.