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1.
The core of a vertex-transitive complementary prism
Marko Orel, 2023, original scientific article

Abstract: The complementary prism $\Gamma \overline{\Gamma}$ is obtained from the union of a graph $\Gamma$ and its complement $\overline{\Gamma}$ where each pair of identical vertices in $\Gamma$ and $\overline{\Gamma}$ is joined by an edge. It generalizes the Petersen graph, which is the complementary prism of the pentagon. The core of a vertex-transitive complementary prism is studied. In particular, it is shown that a vertex-transitive complementary prism $\Gamma \overline{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is a core or its core is a complete graph.
Keywords: graph homomorphism, complementary prism, self-complementary graph, vertex-transitive graph, core
Published in DiRROS: 09.04.2024; Views: 63; Downloads: 33
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2.
Međuodnos svojstava rasta i kvalitete drva na pokusnoj plohi ariša
Mirzeta Memišević Hodžić, Sulejman Sinanović, Dalibor Ballian, 2023, original scientific article

Abstract: Larch (Larix decidua) showed good growth and adaptability to ecological conditions in Bosnia and Herzegovina. The research aims to determine the variability of the morphological traits of larch in the seed stand near Kakanj, the variability at the beginning of heartwood formation, and correlations among these properties.In the larch seed stand, established by seedlings 2+0 in 1979, an area of 0,45 hectares was set aside. The heights and breast height diameters of the trees were on that area were measured in 2002 and 2020, and the basal area and volume were calculated. In 2020, the trees were drilled at breast height with a Presler drill to measure the proportion of heartwood. Average values of all properties for 2002 and 2020 were calculated. The ten, five and one best trees were selected according to the properties of breast height diameter, height, basal area, volume, and the intensities of selection for these traits were calculated. The Pearson correlation coefficient among all investigated traits were calculated. The average breast height diameter of 41-year-old trees was 24,6 cm, average height 21,8 m, basal area 0.0499m2 and volume 0.3745 m3 (table 1), while in 2002. the average breast height diameter was 15,5 cm, height 18,2 m, basal area 0,020 m2, and volume 0,128 m3 (table 1). Between 2002 and 2020, there was an overtaking of trees in breast height diameter and height and thus the base and volume (table 2). The average larch tree starts heartwood formation at the age of 14 years (table 1). The earliest heart-wood formation occurs in the sixth year (tree 495) and the latest in the 32nd year (table 4). Pearson coefficient showed a high positive correlation between breast height diameter and height in 2002 (ta-ble 9), which means that trees with a good height increase also have a good thickness increase. As a result, the correlation between breast height diameter and basal area and breast height diameter and volume, height and basal area, height and volume, and basal area and volume are statistically signifi-cant. Pearson’s correlation coefficient for the year 2020 showed a negative correlation between the be-ginning of heartwood formation and all other traits (table 10), i.e., trees that formed heartwood earlier have higher values of growth traits.The results will be used in the selection processes of larch individuals with good height and thickness growth and early heartwood formation.
Keywords: European larch, morphological properties, core formation, selection
Published in DiRROS: 25.03.2024; Views: 141; Downloads: 113
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3.
The core of a vertex transitive complementary prism of a lexicographic product
Marko Orel, 2023, original scientific article

Abstract: The complementary prism of a graph $\Gamma$ is the graph $\Gamma \overline{\Gamma}$, which is formed from the union of $\Gamma$ and its complement $\overline{\Gamma}$ by adding an edge between each pair of identical vertices in $\Gamma$ and $\overline{\Gamma}$. Vertex-transitive self-complementary graphs provide vertex-transitive complementary prisms. It was recently proved by the author that $\Gamma \overline{\Gamma}$ is a core, i.e. all its endomorphisms are automorphisms, whenever $\Gamma$ is vertex-transitive, self-complementary, and either $\Gamma$ is a core or its core is a complete graph. In this paper the same conclusion is obtained for some other classes of vertex-transitive self-complementary graphs that can be decomposed as a lexicographic product $\Gamma = \Gamma_1 [\Gamma_2]$. In the process some new results aboutthe homomorphisms of a lexicographic product are obtained.
Keywords: graph homomorphism, core, complementary prism, self-complementary graph, vertex-transitive graph, lexicographic product
Published in DiRROS: 19.03.2024; Views: 76; Downloads: 43
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