The Shields-Harary number (SH) is a graph parameter which has been interpreted in terms of network vulnerability. There is no known algorithm for calculating SH. The paper studies the Shields-Harary number for wheel graphs and wheel graphs with deleted outer edges. Wheels with sufficiently many outer edges deleted have the same value of SH as the star. Wheels and wheels with too few edges deleted have a different (but very close) value of SH, which is not precisely known. The paper shows that infinitely many of these graphs have non-integral values of SH. Formerly, only one such graph was known.
