On zero forcing number of graphs and their complements

The zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)\S are colored white) such that V(G) is turned black after finitely many applications of "the color-change rule": a white vertex is converted to a black vertex if it is the only white neighbor of a black vertex. Zero forcing number was introduced and used to bound the minimum rank of graphs by the "AIM Minimum Rank-Special Graphs Work Group". It is known that Z(G) ≥ δ(G), where δ(G) is the minimum degree of G. We show that Z(G) ≤ n - 3 if a connected graph G of order n has a connected complement graph [Formula: see text]. Further, we characterize a tree or a unicyclic graph G which satisfies either [Formula: see text] or [Formula: see text].

NEW SPECIES OF HEMEROBIUS

Preparatory to a revision of the Nearctic Hemerobiidæ I present descriptions of a few new species of Hemerobius. A few of the names have been used already elsewhere, but without description. Outline figures of the male genitalia will be given in the forthcoming revision.Hemerobius transversus, n. sp.—Face shining black, vertex and antennæ pale yellowish; thorax pale, a black stripe across front part of the mesothorax; abdomen brownish; legs pale yellowish. Wings with the margins faintly but broadly clouded with brown; the gradate series marked with dark brown, and a brown band between the first and second first.

NEW APOIDEA FROM MONTANA

Bombus Cooleyi, n. sp.— ♀. Length, 16–17 mm. Integument black; clothing black, yellowish white, pale ochreous yellow and rusty yellow. Head, seen from in front, about as wide as long. Malar space abont one-sixth the length of eye. Third segment of antenna one-half longer than fourth, and scarcely longer than fifth. Face thickly clothed with pale yellowish-white hair, on the sides mixed with black. Vertex clothed with yellowish-white hair, which is fringed in front with black. Cheeks clothed with brownish-black, sometimes slightly mixed with whitish, hair. Clypeus shining, sparsely punctured, labrum fringed on free edge with rusty yellorv hair. Clothing of thorax above and on sides yellowish white, mixed with black in front of insertion of wings. A broad patch of black berween the wings surrounds the smooth, polished mesothoracic disc and extends back in a point over the middle of metathorax. On each side of metathorax is a tuft of yellowish white bair.

DESCRIPTION OF NEW GENERA AND SPECIES OF THE GEOMETRINA OF NORTH AMERICA

Leptomeris nigrodiscalis, n. sp.Expands 24 mm. Palpi and front black; vertex ochre-white; antennæ whitish below, blackish or smoky above; thorax and abdomen white, slightly ochre stained. Fore wings white, with four somewhat indistinct, broad, even, wavy, ochre lines, the first well out from base, the second just outside of discal spot, the third in outer space, the fourth marginal; hind wings with corresponding broad lines; discal points prominent, rather large, jet black; four fine marginal black points below apex on fore wings; beneath more diffuse, more fuscous, less ochreous, the lines less determinate; discal spots not so marked, black points as above on margins of fore wings, and some appearing below apex along margin on hind wings.

A NEW SPECIES OF PROTANDRENA, CKLL.

Protandrena Cockerelli, n. sp.—♀. Length, 12 mm.; shining black. Upper half of clypeus, lower portion of supraclypeal area, and part of side pieces, bright yellow, all forming a band across the face one-half broader than high, and of equal breadth throughout; knees yellow spotted. Head rounded, broader than high, and covered with a short growth of gray hair, longer on cheeks and thickest at bave of antennæ; lower haif of clypeus and two small dots near lower edge of band black, not deeply or closely punctured; antennæ black at base, becoming brown towards the tip; first joint of flagellum not quite as long as the second and third combined; mandibles black; vertex deeply but not very closely punctate.