Plant materials belonging to the natural populations of T. cilicius were collected from Sandras Mountain (Muğla, Turkey, Fig. 1A). The legal authorization letter for sample collection was obtained from Mugla Metropolitan Municipality, Department of Agricultural Services (Doc. number: 10452259-622.03-E.930/15708). Surface sterilization of T. cilicicus shoot tips was performed according to surface sterilization protocol for Thymus spp. developed by Ozudogru et al. (2011). The shoots (~1 cm) were washed under tap water for half an hour, and subsequently, they were treated with 70% ethanol for 5 min, 3.5% commercial bleach (Domestos®) for 15 min, then they were rinsed in distilled water at least three times. After surface sterilization, the shoot tips (~0.1 cm) were excised and transferred to semi-solid MS medium supplemented with 1 mg L^-1 6-benzylaminopurine (BA), 20 g L^-1 sucrose and 3 g L^-1 phytagel.

The shoot apical meristems (~0,5 - 2 mm in size) were isolated from in vitro grown T. cilicicus shoots (culture stating material, Fig. 1B-D) on semi-solid MS medium described above and they were transferred to MS or OM media (Rugini 1993) supplemented with different combinations of salt and/or plant growth regulators contents (Tab. 1). All cultures were incubated in a growth room under the standard culture conditions [25 ± 2 °C, 16-h photoperiod with cool daylight fluorescent lamps (50 μmol m⁻² s⁻¹)]. These apical meristems were isolated by eight different persons to decrease standard errors of manual ability. Ten apical meristems were used for each medium combination and each treatment was repeated at least three times.

The data of the in vitro meristem regenerations on different media previously described before were collected after four weeks. The averages of regeneration percentages, shoot numbers and shoot lengths were calculated with standard errors via IBM SPSS (V22.0) statistical program. The significant treatment differences were selected by a non-parametric statistical test, and the post hoc multiple comparisons test (Marascuilo and McSweeney 1977). Discrete data were subjected to ANOVA, followed by the least significant difference test at P≤0.05 to compare means (homology between values of regeneration percentages, shoot number and shoot length avarages were evaluated seperately and indicated different letters.

The graph kernel statistical index formula was created to determine the optimum medium composition taking into account all parameters for the best in vitro propagation of T. cilicicus (This method was used for the first time for in vitro propagation data analysis).

Graphs are natural mathematical concepts used in order to express structured data sets. In mathematics, a graph can be expressed via a tuple where is the set of vertices (or nodes) and is a set of edges (or links) which is the subset of . In a graph structure, the information of the components of a combinatoric system is encoded to vertices and the information of the relations between vertices is encoded to edges. If the edges are symmetrical, that is, if implies , then is called directed, otherwise undirected. Moreover, if , then is called a simple graph. Throughout this study, we will assume that a graph is simple and undirected. The vertices connected via an edge in are called adjacent.

In many daily life applications, a non-negative value called weight is assigned to each edge of . Edge weights can be determined based on metric distances, costs, or similarities. Networks modeled with edge-weighted graphs are widely used in genomics and system biology. There are only a few studies on vertex-weighted graphs as compared to edge-weighted network analysis. Like edge-weighted graphs, a vertex-weighted graph can be defined by assigning a non-negative value to each vertex of the graph (Knisley and Knisley, 2014).

Let be a vertex-weighted graph with the weight function and . The Laplacian matrix of is defined by the entities

where . Similar to the Laplacian operator defined in continuous spaces, the Laplacian operator defined on measures how a function's value in a vertex differs from vertices adjacent to .

In this study, for the comparison of vertex-weighted graphs, a graph kernel based on the Laplacian matrix described above is defined for the first time. The Laplacian graph kernel we defined created a hierarchy of nested subgraphs, enabling comparison of the graphs. Let and be two vertex-weighted graphs and their Laplacian matrices are and , respectively. Then the kernel function we use is

where and for the identity matrix , a parameter and is Moore-Penrose inverse of a matrix. The kernel function we defined is symmetrical and positive semi-definite. The Laplacian graph kernel we present captures the similarity between the general structures of the two graphs. However, this assumes that both graphs have the same size and are invariant in the permutations of the vertices.

While representing each experimental group with graphs, graphs with 30 vertices were used because there were 10 meristems in each petri and 3 different petri dishes. The vertices symbolizing the meristems from which shoots developed are related with each other with an edge and vertex weighted graphs are obtained. Vertex weighting function is defined as

where y is the number of dead shoots. Each case of the experiments conducted in this study is shown with vertex-weighted simple graphs. Graphs are effective mathematical structures for the analysis of data structures with many heterogeneity levels. For this reason, they encode a variety of relationships that classical statistical methods cannot capture. In the graph comparison apporach, each graph obtained in this study is compared using the graph kernel function, which models the best case. The graph that models the best case is a complete graph, that is, there is a link between each node. Also, since a weight is determined at the vertices of the graph, this complete graph model takes the maximum value within the weights. The complete graph model, with each vertex weight being the maximum of all cases, represents the best case, and the graphs obtained empirically are compared with the best case with the aid of the kernel function.

The DNA samples isolated from the mother plant leaves (culture-starting material) and the shoots grown on the eight optimal media (according to graph kernel multiple propagation index, each regenerated shoot on each medium was analysed as individually with 24 primers) were compared to determine genetic stability using 24 ISSR primers (Kaya 2015). The DNAs were isolated by using Doyle and Doyle’s (1987) CTAB protocol. Polymerase chain reactions were performed in a 25 μL reaction mixture containing 40 ng DNA, 0.4 mM primer, 0.4 mM of each dNTP, 2.5 mM MgCl₂ and 1 unit Taq DNA polymerase. PCR conditions were as follows: after 3 min denaturation at 95 °C, the reactions were continued for 35 cycles of 15 sec at 95 °C; 30 sec at 54 °C; 3 min at 68 °C followed by a 10 min lag at 72 °C. PCR products were separated on 1.5% agarose gel at 80 V and were visualized under UV light via staining with 0.5 μg ml^-1 ethidium bromide solution. The band profiles were scored as 1 (present) or 0 (absent) and all data were analyzed for the determination of genetic stability (Kaya and Souza 2017).

The OM medium had a more positive effect on multiple propagation than MS medium and a reasonable explanation for this result is the reduced NO₃^- (NH₄NO₃, KNO₃) salts as nitrogen sources in OM medium (Arab et al. 2014). At the same time, OM medium also contains a different nitrogen salt [Ca(NO₃)₂, 2.54 mM] as a nitrogen source. Many works to determine the potential utilities of different nitrogen sources such as NO₃^- and NH₄⁺ have been undertaken and their results indicated that the different concentrations of different forms of nitrogen source in the nutrient media have produced very positive responses on somatic embryo development (Leljak-Levanić et al. 2004), plant recovery efficiency in ovule cultures (McCoy and Smith 1986) and shoot regeneration (Vinterhalter et al. 2007).

Adenine type cytokinin plant growth regulators such as BA and KIN play different roles in many aspects of plant growth, development, anabolic and catabolic stimulation processes in plant cell metabolism (Quadri et al. 2012). Kinetin is beneficial for inducing cell development, proliferation and new shoot formation when supplemented to the culture medium (Castilho et al. 2019), and it has also been suggested for micropropagation of different Thymus species, for example, T. vulgaris, T. longicaulis (Ozudogru et al. 2011), T. cariensis (Ozudogru and Kaya 2012) and T. hyemalis (Nordine et al. 2013), T. persicus (Bakhtiar et al. 2016).

Charcoal effects on in vitro regeneration of plant tissues depend on the charcoal type, their activation type and also the plant species being cultured. In vitro culture medium supplemented with activated charcoal can have either an adverse or an advantageous effect on tissue growth and development and this depends on plant tissue, plant species, medium content and the aim of the study. The activated charcoal addition to in vitro culture media could have different effects in in vitro cultures, such as the provison of a degree of darkness, inhibition of undesirable substances, and plant growth regulator adsorption. A problem faced in tissue culture studies, especially with plant species containing phenolic compounds during their culture initial phase, is the browning of tissues and eventual death of the explants because of excessive polyphenol production caused by plant defense reactions. These phenolic compounds mostly have been known as being inhibitory or undesirable substances that should be eliminated from or avoided in in vitro culture conditions. Since activated charcoal adsorbs phenolic compounds, discoloration is prevented and polyphenol oxidase and peroxidase are renderedinactive. It also decreases browning of tissues and culture media, thus viability and regeneration of tissues increase (Pan and Staden 1998).

The effects of Fe-EDDHA, AgNO₃ and H₃BO₃ were tested for T. cilicicus meristem regeneration on MS and OM media containing BA or KIN (with or without charcoal) combinations (Tab. 1). For plant cells, one of the most essential microelements that carry out metabolic pathways such as chlorophyll biosynthesis is iron. Therefore, iron insufficiency restricts plant growth and development and also stimulates interveinal chlorosis (Guerinot 2001). Metabolic pathways such as photosynthesis, respiration, DNA, RNA and protein biosynthesis, detoxification and nitrogen fixation is usually based on iron redox enzymes including ferredoxin, cytochromes, ribonucleotide reductase, lipoxygenase, oxidases, catalases, superoxide dismutases, nitrite and nitrate reductases respectively (Curie et al. 2009).

Ethylene accumulation in in vitro culture vessels may cause growth inhibition, some abnormal plant forms, leaf senescence and leaf reduction in micropropagated plant cultures (Steinitz et al. 2010), but these effects generally depend on plant species, culture type, explant type and even vessel type (Jha et al. 2007). Since ethylene acts as an inhibitor for in vitro cultures, AgNO₃ improves the growth and regeneration of different plant types when it is added to the culture medium (Sandra and Maira, 2013).

H₃BO₃ is generally used as a boron source for in vitro cultures and its defiency may cause different abnormalities such as the inhibition of metabolic activities and/or alteration of plant morphology. The possible function of Boron is the production of cell wall compounds such as uracil-diphosphate glucose, glucose-1phosphate and 6-phosphate gluconate (Matoh 1997).

In the current work, although the Fe-EDDHA had a more benefical effect on meristem regeneration than AgNO₃ and H₃BO₃ (Fig. 4D), none of the medium combinations containing Fe-EDDHA, AgNO₃ or H₃BO₃ was the optimum medium composition. These results almost resembled previous works on in vitro regeneration of Lagenaria siceraria (Saha et al. 2007), Cucurbita maxima (Lee et al. 2003), T. vulgaris (Ozudogru et al. 2011).

To determine the somaclonal variation, ISSR marker systems have been successfully used on some in vitro propagated plant species such as thyme (Ozudogru et al 2011), grapevine (Nookaraju and Agrawal 2012), sugarcane (Kaya and Souza, 2017). The previous reports support the current study that the plantlets derived from T. cilicicus organized meristems can be genetically stable after treatment under in vitro conditions. The ISSR analysis of in vitro grown T. cilicicus revealed no somaclonal variation among seedlings similar to in vitro propagated T. vulgaris and T. longicaulis (Ozudogru et al. 2011).

A micropropagation system that supports genetically stable clonal multiplication for T. cilicicus has been determined using the graph kernel statistical method. The results of the current study showed that the semi-solid MS medium supplemented with 20 g L^-1 sucrose, 1 mg L^-1 KIN and 10 g L^-1 charcoal was the best for the proliferation of T. cilicicus meristems. Furthermore, the micropropagated plants were successfully rooted and acclimatized to greenhouse conditions. In this study, we present a graph kernel multiple propagation index that considers all possible parameters for determining an optimum proliferation medium. Such an index is used for the first time for the determination of the optimum proliferation medium. Moreover, such a kernel function can be applied in several subjects. This study of the micropropagation of T. cilicicus, an aromatic-medicinal plant species, indicates that in vitro propagation is feasible and practicable for rapid multiplication of economically important plant species, faster introduction of new cultivars with beneficial properties, and for rapid clonal propagation of healthy, bacterium-, fungus- and virus-free, healthy and genetic stable plant material.