Let G be a connected graph with vertex set V(G) and edge set E(G). The Sombor index of G is defined asRecent results of Sombor index can be found in References [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].A chemical tree is a tree with d v ≤ 4 for all v ∈ V(G). For details on chemical trees, see . Let T n (CT n ) be the set of trees (chemical trees) with n vertices. Let T n,k (CT n,k ) be the set of trees (chemical trees) with n vertices and k pendent vertices.
Cosmetics have a long history in China, but their origins remain unclear. Cosmetic industry potentially originated in the , but little is known about the early manufacture and use of cosmetics. The Liujiawa site, located on the southern edge of the Loess Plateau in northern China, was the late capital of the ancient Rui state in the early to mid-Spring and Autumn Period. During the excavation, a sealed small and exquisite container with suspected cosmetic use was unearthed from tomb M49 belonging to a male associated with the aristocratic class. We report in this paper the multidisciplinary application of ATR-FTIR, XRD, SEM, stable isotope analysis, GC-MS and GC-C-IRMS analysis of the residue inside the container, demonstrating that the residue, which was made of ruminant adipose fat mixed with monohydrocalcite from cave moonmilk, was likely used as cosmetic face cream by the nobleman of the Rui state. This study provides an early example of cosmetic production in China, and, together with the prevalence of similar cosmetic containers during this period,
For a connected graph [Formula: see text], the Mostar index is defined as [Formula: see text], where [Formula: see text] (respectively, [Formula: see text]) is the number of vertices of [Formula: see text] closer to [Formula: see text] (respectively, [Formula: see text]) than [Formula: see text] (respectively, [Formula: see text]). In this paper, we determine the first five maximal (respectively, the first five minimal) values of the Mostar index among all phenylene chains with [Formula: see text] hexagons, the corresponding extremal chains are completely characterized, respectively.
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