Let Ŵ = (V, E) be a connected graph. A vertex i ∈ V recognizes two elements (vertices or edges) j, k ∈ E ∩ V, if d Ŵ (i, j) = d Ŵ (i, k). A set S of vertices in a connected graph Ŵ is a mixed metric generator for Ŵ if every two distinct elements (vertices or edges) of Ŵ are recognized by some vertex of S. The smallest cardinality of a mixed metric generator for Ŵ is called the mixed metric dimension and is denoted by β m. In this paper, the mixed metric dimension of a generalized Petersen graph P(n, 2) is calculated. We established that a generalized Petersen graph P(n, 2) has a mixed metric dimension equivalent to 4 for n ≡ 0, 2(mod4), and, for n ≡ 1, 3(mod4), the mixed metric dimension is 5. We thus determine that each graph of the family of a generalized Petersen graph P(n, 2) has a constant mixed metric dimension.

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A vertex k ∈ V G determined two elements (vertices or edges) , m ∈ V G ∪ E G , if d G (k,) = d G (k, m). A set R m of vertices in a graph G is a mixed metric generator for G, if two distinct elements (vertices or edges) are determined by some vertex set of R m. The least number of elements in the vertex set of R m is known as mixed metric dimension, and denoted as dim m (G). In this paper, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are P 2 n the square of a path, T (P n) total graph of a path, the middle graph of a path M (P n), and splitting graph of a path S(P n). We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path S(P n). INDEX TERMS Mixed metric dimension, Metric dimension, Edge metric dimension, Path related graphs.

Bovine lactoferrin (bLf) is a multifunctional glycoprotein and a good candidate for producing diverse bioactive peptides, which are easily lost during over-digestion. Accordingly, the effects of laccase-mediated pectin–ferulic acid conjugate (PF) and transglutaminase (TG) on improving the production of bLf active peptides by in vitro gastrointestinal digestion were investigated. Using ultra-high-performance liquid chromatography tandem mass spectroscopy (UPLC-MS-MS), the digests of bLf alone, PF-encapsulated bLf complex (LfPF), and TG-treated LfPF complex (LfPFTG) produced by conditioned in vitro gastric digestion (2000 U/mL pepsin, pH 3.0, 37 °C, 2 h) were identified with seven groups of active peptide-related fragments, including three common peptides (VFEAGRDPYKLRPVAAE, FENLPEKADRDQYEL, and VLRPTEGYL) and four differential peptides (GILRPYLSWTE, ARSVDGKEDLIWKL, YLGSRYLT, and FKSETKNLL). The gastric digest of LfPF contained more diverse and abundant detectable peptides of longer lengths than those of bLf and LfPFTG. After further in vitro intestinal digestion, two active peptide-related fragments (FEAGRDPYK and FENLPEKADRDQYE) remained in the final digest of LfPFTG; one (EAGRDPYKLRPVA) remained in that of bLf alone, but none remained in that of LfPF. Conclusively, PF encapsulation enhanced the production of bLf active peptide fragments under the in vitro gastric digestion applied. TG treatment facilitated active peptide FENLPEKADRDQYE being kept in the final gastrointestinal digest.

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