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2. The extremal generalised Randić index for a given degree rangeJohn Haslegrave, 2025, izvirni znanstveni članek Opis: O and Shi proved that the Randić index of any graph G with minimum degree at least δ and maximum degree at most Δ is at least sqrt(δΔ)/(δ+Δ) |G|, with equality if and only if the graph is (δ, Δ)-biregular. In this note we give a short proof via a more general statement. As an application of our more general result, we classify for any given degree range which graphs minimise (or maximise) the generalised Randić index for any exponent, and describe the transitions between different types of behaviour precisely.
Ključne besede: Randić index, bounded-degree graph, extremal problem
Objavljeno v RUP: 03.11.2025; Ogledov: 969; Prenosov: 8
 Celotno besedilo (403,78 KB) |
3. On regular graphs with Šoltés verticesNino Bašić, Martin Knor, Riste Škrekovski, 2025, izvirni znanstveni članek Opis: 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 RUP: 10.09.2025; Ogledov: 1363; Prenosov: 11
Celotno besedilo (456,75 KB) |
4. Selected topics on Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2024, izvirni znanstveni članek Ključne besede: graph distance, Wiener index, average distance, topological index, molecular descriptor, chemical graph theory
Objavljeno v RUP: 26.05.2025; Ogledov: 1680; Prenosov: 12
Celotno besedilo (516,10 KB) |
5. Exploring the bounds and relationships of chemical graph indices : master's thesisArbër Avdullahu, 2023, magistrsko delo Ključne besede: energy of a graph, Randić index, first Zagreb index, IRB index, extremal graph, spectrum, eigenvalue, metaheuristic
Objavljeno v RUP: 05.10.2023; Ogledov: 2587; Prenosov: 53
Celotno besedilo (1,02 MB) |
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7. Mathematical aspects of Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2016, izvirni znanstveni članek Opis: The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
Ključne besede: Wiener index, total distance, topological index, molecular descriptor, chemical graph theory
Objavljeno v RUP: 03.01.2022; Ogledov: 5432; Prenosov: 57
Celotno besedilo (434,58 KB) |
8. Edge-contributions of some topological indices and arboreality of molecular graphsTomaž Pisanski, Janez Žerovnik, 2009, izvirni znanstveni članek Opis: Some graph invariants can be computed by summing certain values, called edge-contributions over all edges of graphs. In this note we use edge-contributions to study relationships among three graph invariants, also known as topological indices in mathematical chemistry: Wiener index, Szeged index and recently introduced revised Szeged index. We also use the quotient between the Wiener index and the revised Szeged index to study tree-likeness of graphs.
Ključne besede: mathematical chemistry, chemical graph theory, topological index, revised Szeged index
Objavljeno v RUP: 30.12.2021; Ogledov: 4280; Prenosov: 34
Celotno besedilo (158,93 KB) |
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10. Connected graphs of fixed order and size with minimal index : structural considerationsSlobodan Simić, Enzo M. Li Marzi, Francesco Belardo, 2004, izvirni znanstveni članek Ključne besede: spekter grafa, največja lastna vrednost, spektralni radij, indeks grafa, graph spectrum, largest eingenvalue, spectral radius, graph index, nested split graphs
Objavljeno v RUP: 15.10.2015; Ogledov: 6225; Prenosov: 63
 Povezava na celotno besedilo |
