Sandi Klavžar

Bio: Sandi Klavžar is an academic researcher from University of Ljubljana. The author has contributed to research in topics: Domination analysis & Vertex (geometry). The author has an hindex of 36, co-authored 307 publications receiving 5322 citations. Previous affiliations of Sandi Klavžar include University of Maribor & Mechanics' Institute.

Wiener Index of Hexagonal Systems

Abstract: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. Hexagonal systems (HS's) are a special type of plane graphs in which all faces are bounded by hexagons. These provide a graph representation of benzenoid hydrocarbons and thus find applications in chemistry. The paper outlines the results known for W of the HS: method for computation of W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's. A few open problems are mentioned. The chemical applications of the results presented are explained in detail.

Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments

Abstract: Protein structures evolved through a complex interplay of cooperative interactions, and it is still very challenging to design new protein folds de novo. Here we present a strategy to design self-assembling polypeptide nanostructured polyhedra based on modularization using orthogonal dimerizing segments. We designed and experimentally demonstrated the formation of the tetrahedron that self-assembles from a single polypeptide chain comprising 12 concatenated coiled coil-forming segments separated by flexible peptide hinges. The path of the polypeptide chain is guided by a defined order of segments that traverse each of the six edges of the tetrahedron exactly twice, forming coiled-coil dimers with their corresponding partners. The coincidence of the polypeptide termini in the same vertex is demonstrated by reconstituting a split fluorescent protein in the polypeptide with the correct tetrahedral topology. Polypeptides with a deleted or scrambled segment order fail to self-assemble correctly. This design platform provides a foundation for constructing new topological polypeptide folds based on the set of orthogonal interacting polypeptide segments.

Domination Game and an Imagination Strategy

Abstract: The domination game played on a graph $G$ consists of two players, Dominator and Staller, who alternate taking turns choosing a vertex from $G$ such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay the process as much as possible. The game domination number $\gamma_g(G)$ (resp., $\gamma_g'(G)$) is the number of vertices chosen when Dominator (resp., Staller) starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph $G$, $\gamma(G)\leq\gamma_g(G)\leq2\gamma(G)-1$, and that all possible values can be realized. It is proved that for any graph $G$, $\gamma_g(G)-1\leq\gamma'_g(G)\leq\gamma_g(G)+2$, and that most of the possibilities for mutual values of $\gamma_g(G)$ and $\gamma_g'(G)$ can be realized. A connection with Vizing's conjecture is established, and a lower bound on the game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

Graphs S(n, k) and a Variant of the Tower of Hanoi Problem

Abstract: For any n ≥ 1 and any k ≥ 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S(n, 3) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S(n, k). Together with a formula for the distance, this result is used to compute the distance between two vertices in O(n) time. It is also shown that for k ≥ 3, the graphs S(n, k) are Hamiltonian.

Graph Coloring Problems

Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

Wiener Index of Trees: Theory and Applications

Abstract: The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.