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Povzetek: A function $f$ that assigns values from the set $\{0, 1, 2\}$ to each vertex of a graph is called a $2$-rainbow independent dominating function, if the vertices assigned the value $1$ form an independent set, the vertices assigned the value $2$ form another independent set, and every vertex to which $0$ is assigned has at least one neighbor in each of the mentioned independent sets. The weight of this function is the total number of vertices assigned nonzero values. The $2$-rainbow independent domination number of $G$, $\gamma_{{\rm ri}2}(G)$, is the minimum weight of such a function. Motivated by a real-life application, we study the $2$-rainbow independent domination number of the complementary prism $G \overline{G}$ of a graph $G$, which is constructed by taking $G$ and its complement $\overline{G}$, and then adding edges between corresponding vertices. We provide tight bounds for $\gamma_{{\rm ri}2}(G \overline{G})$, and characterize graphs for which the lower bound, i.e. $\max \{ \gamma_{{\rm ri}2}(G), \gamma_{{\rm ri}2}(\overline{G})\}+1$, is attained. The obtained results can, in practice, enable the prediction of the cost estimate for a given communication or surveillance network.
Ključne besede: graph theory, domination, 2-rainbow independent domination, complementary prism
Objavljeno v DiRROS: 28.04.2025; Ogledov: 10; Prenosov: 5
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Povzetek: Let $T$ be an operator on Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \\ 0 & -I_2 \end{matrix} \right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1, 2$). Furthermore, $T$ has necessarily the form $T = \left(\begin{matrix} 0 & * \\ * & 0 \end{matrix} \right) $ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P = \left(\begin{matrix} I_1 & 0 \\ 0 & 0 \end{matrix} \right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1 \le p \le \infty$) are commutators of idempotent operators.
Ključne besede: Banach spaces, operators, idempotents, commutators
Objavljeno v DiRROS: 24.04.2025; Ogledov: 51; Prenosov: 24
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Povzetek: Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a Šoltés vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, Šoltés posed the problem of identifying all connected graphs ▫$G$▫ with the property that all vertices of $G$ are Šoltés vertices. The only such graph known to this day is $C_{11}$. As the original problem appears to be too challenging, several relaxations were studied: one may look for graphs with at least $k$ Šoltés vertices; or one may look for $\alpha$-Šoltés graphs, i.e. graphs where the ratio between the number of Šoltés vertices and the order of the graph is at least $\alpha$. Note that the original problem is, in fact, to find all $1$-Šoltés graphs. We intuitively believe that every $1$-Šoltés graph has to be regular and has to possess a high degree of symmetry. Therefore, we are interested in regular graphs that contain one or more Šoltés vertices. In this paper, we present several partial results. For every $r\ge 1$ we describe a construction of an infinite family of cubic $2$-connected graphs with at least $2^r$ Šoltés vertices. Moreover, we report that a computer search on publicly available collections of vertex-transitive graphs did not reveal any $1$-Šoltés graph. We are only able to provide examples of large $\frac{1}{3}$-Šoltés graphs that are obtained by truncating certain cubic vertex-transitive graphs. This leads us to believe that no $1$-Šoltés graph other than $C_{11}$ exists.
Ključne besede: Šoltés problem, Wiener index, regular graphs, cubic graphs, Cayley graph, Šoltés vertex
Objavljeno v DiRROS: 17.04.2025; Ogledov: 109; Prenosov: 48
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Povzetek: A subset $S$ of vertices of a graph $G$ is a general position set if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position number ${\rm gp}^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.
Ključne besede: general position number, geodetic number, universal line, computational complexity, Kneser graphs, line graphs
Objavljeno v DiRROS: 11.04.2025; Ogledov: 120; Prenosov: 61
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Povzetek: Let $L$ be a Lie algebra and let $A$ be an associative commutative algebra with unity, both over the same field $F$. We consider the following question. Is every generalized derivation (resp. quasiderivation) of $L \otimes A$ the sum of a derivation and a map from the centroid of $L \otimes A$, if the same holds true for $L$?
Ključne besede: current Lie algebras, derivation, generalized derivation, Lie algebras, quasiderivation, tensor product of algebras
Objavljeno v DiRROS: 31.03.2025; Ogledov: 160; Prenosov: 89
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