THE SHIELDS-HARARY NUMBER FOR WHEEL AND BROKEN WHEEL GRAPHS

The Shields-Harary number (SH) is a graph parameter which has been int erpreted in terms of network vulnerability. There is no known algorith m for calculating SH. In this paper, we study the Shields-Harary numbe r for wheel graphs and wheel graphs with deleted outer edges. Wheels w ith sufficiently many outer edges deleted have the same value of SH as the star. Wheels and wheels with too few edges deleted have a differe nt (but very close) value of SH, which is not precisely known. We show that infinitely many of these graphs have non-integral values of SH. Formerly, only one such graph was known.