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Povzetek: In this paper we introduce a product-like operation that generalizes the construction of the generalized Sierpiński graphs. Let $G$, $H$ be graphs and let $f: V(G) \to V(H)$ be a function. Then the Sierpiński product of graphs $G$ and $H$ with respect to $f$, denoted by $G\otimes_f H$, is defined as the graph on the vertex set $V(G) \times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $\{g, g'\}$ of $G$ there is an edge between copies $gH$ and $g'H$ of form $\{(g, f(g'), (g', f(g))\}$. Some basic properties of the Sierpiński product are presented. In particular, we show that the graph $G\otimes_f H$ is connected if and only if both graphs $G$ and $H$ are connected and we present some conditions that $G, \, H$ must fulfill for $G\otimes_f H$ to be planar. As for symmetry properties, we show which automorphisms of $G$ and $H$ extend to automorphisms of $G\otimes_f H$. In several cases we can also describe the whole automorphism group of the graph $G\otimes_f H$. Finally, we show how to extend the Sierpiński product to multiple factors in a natural way. By applying this operation $n$ times to the same graph we obtain an alternative approach to the well-known $n$-th generalized Sierpiński graph.
Ključne besede: Sierpiński graphs, graph products, connectivity, planarity, symmetry
Objavljeno v DiRROS: 19.03.2024; Ogledov: 133; Prenosov: 50
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Povzetek: In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of $Q_d$ differ by at most one for every integer $d$. In particular, if $d$ is odd, then the metric and the edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube $Q_d$ are equal for every $d \ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when d is large enough.
Ključne besede: edge metric dimension, mixed metric dimension, metric dimension, hypercubes
Objavljeno v DiRROS: 19.03.2024; Ogledov: 115; Prenosov: 34
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Objavljeno v DiRROS: 19.03.2024; Ogledov: 96; Prenosov: 47
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Povzetek: Introduction. CD133 is a marker for a population of glioblastoma (GBM) and normal neural stem cells (NNSC). We aimed to reveal whether the migratory potential and differentiation of these stem cells is associated with CD133 expression and with cathepsin proteases (Cats). Materials and methods. The invasiveness of normal NNSC, GBM/CD133+ cell lines and GBM spheroids was evaluated in 3D collagen, as well as of U87-MG and normal astrocytes (NHA) grown in monolayers in 2D Matrigel. Expression of Cats B, L and S was measuredat mRNA and activity levels and their relation to invasiveness, to CD133 mRNA in 26 gliomas, and to the survival of these patients. Results. The average yield of CD133+ cells from GBM samples was 9.6%. Survival of patients with higher CD133 mRNA expression was significantly shorter (p< 0.005). Invasion, associated with proteolytic degradation of matrix, was higher in normal stem cells and GBM spheroids and cells than in isolated GBM CD133+ cells. In glioma samples, there was no correlation between CD133 mRNA expression and Cat mRNAs, but there was an inverse correlation with Cat activities. Conclusions. The study confirms CD133 as a prognostic marker for the survival of GBM patients. We demonstrated that NNSC have higher invasion potential and invade the collagen matrix in a mode different from that of GBM,initiating stem cell spheres. This result could have implications for the design of new therapeutics, including protease inhibitors that specifically target invasive tumour stem cells. Increased activity of cathepsins in CD133- cells suggests their role in the invasive behaviour of GBM.
Objavljeno v DiRROS: 19.03.2024; Ogledov: 127; Prenosov: 33
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Povzetek: Background. Triple negative breast cancer (TNBC) is defined by a lack of expression of both estrogen (ER) and progesteron(PgR) receptors as well as human epidermal growth factor receptor 2 (HER2). Our retrospective analysis addressed prognostic factors for short- and long-term outcomes of patients (pts) with TNBC pts treated in routine clinical practice. Patient and methods.Our retrospective study included 269 TNBC treated at Institute of Oncology Ljubljana between March 2000 and December 2006. The collected data included patientsć, tumoursć and treatmentsć characteristics. The survival analyses were performed using the Kaplan-Meier method. The Cox proportional hazard model was used in the multivariate analysis. Results. The median age ofour patients was 55.3 yrs (23-88.5) and the median follow-up was 5.9 yrs (0.3-9.6). Six (2%) pts experienced local only, 79 (92%) pts distal recurrenceand 66 (24%) died. The predominant localisation of the first relapsewas in visceral organs (70.4%). The 5-year disease-free survival (DFS) for the entire group was 68.2% and the 5-year overall survival (OS) was 74.5%.We found a pattern of high recurrence rate in the first 3 years following the diagnosis and a clear decline in recurrence rate over the next 3years. In the univariate analysis age, nodal status, size and lymphovascular invasion (LVI) were found to have a significant impact on DFS as well as on OS. In the multivariate analysis only age (HR=1.79; 95%CI=1.14-2.82; p=0.012) and nodal status (HR=2.71; 95%CI=1.64-4.46; p<0.001) retained their independent prognostic value for DFS and for OS only the nodal status (HR=2.96; 95%CI=1.51-5.82; p=0.002). (Abstract truncated at 2000 characters)
Objavljeno v DiRROS: 19.03.2024; Ogledov: 109; Prenosov: 33
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