Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Eigenvalues, traceability, and perfect matching of a graph

It is well known that the recognition problem of the existence of a perfect matching in a graph, as well as the recognition problem of its Hamiltonicity and traceability, is NP-complete. Quite recently, lower bounds for the size and the spectral radius of a graph that guarantee the existence of a perfect matching in it have been obtained. We improve these bounds, firstly, by using the available bounds for the size of the graph for existence of a Hamiltonian path in it, and secondly, by finding new lower bounds for the spectral radius of the graph that are sufficient for the traceability property. Moreover, we develop the recognition algorithm of the existence of a perfect matching in a graph. This algorithm uses the concept of a (κ,τ)-regular set, which becomes polynomial in the class of graphs with a fixed cyclomatic number.

Folds induced by multiple parallel or antiparallel double-helices: (pseudo)knotting of single-stranded RNA

ABSTRACTWe develop tools to explore and catalogue the topologies of knotted or pseudoknotted circular folds due to secondary and tertiary interactions within a closed loop of RNA which generate multiple double-helices due (for example) to strand complementarity. The fold topology is captured by a ‘contracted fold’ which merges helices separated by bulges and removes hairpin loops. Contracted folds are either trivial or pseudoknotted. Strand folding is characterised by a rigid-vertex ‘polarised strand graph’, whose vertices correspond to double-helices and edges correspond to strands joining those helices. Each vertex has a plumbline whose polarisation direction defines the helical axis. That polarised graph has a corresponding circular ribbon diagram and canonical alphanumeric fold label. Key features of the ‘fully-flagged’ fold are the arrangement of complementary domains along the strand, described by a numerical bare fold label, and a pair of binary ‘flags’: a parity flag that specifies the twist in each helix (even or odd half-twists), and an orientation flag that characterises each double-helix as parallel or antiparallel. A simple algorithm is presented to translate an arbitrary fold label into a polarised strand graph. Any embedding of the graph in 3-space is an admissible fold geometry; the simplest embeddings minimise the number of edge-crossings in a planar graph drawing. If that number is zero, the fold lies in one of two classes: (a)-type ‘relaxed’ folds, which contain conventional junctions and (b)-type folds whose junctions are described as meso-junctions in H. Wang and N.C. Seeman, Biochem, vol. 34, pp920-929. (c)-type folds induce polarised strand graphs with edge-crossings, regardless of the planar graph drawing. Canonical fold labelling allows us to sort and enumerate all ‘semi-flagged’ folds with up to six contracted double-helices as windings around the edges of a graph-like fold skeleton, whose cyclomatic number - the ‘fold genus’ - ranges from 0 – 3, resulting in a pair of duplexed strands along each skeletal edge. Those semi-flagged folds admit both even and odd double-helical twists. Appending specific parity flags to those semi-flagged folds gives fully-flagged (a)-type folds, which are also enumerated up to genus-3 cases. We focus on all-antiparallel folds, characteristic of conventional ssRNA and enumerate all distinct (a), (b) and (c)-type folds with up to five double-helices. Those circular folds lead to pseudoknotted folds for linear ssRNA strands. We describe all linear folds derived from (a) or (b)-type circular folds with up to four contracted double-helices, whose simplest cases correspond to so-called H, K and L pseudoknotted folds, detected in ssRNA. Fold knotting is explored in detail, via constructions of so-called antifolds and isomorphic folds. We also tabulate fold knottings for (a) and (b)-type folds whose embeddings minimise the number of edge-crossings and outline the procedure for (c)-type folds. The inverse construction - from a specific knot to a suitable nucleotide sequence - results in a hierarchy of knots. A number of specific alternating knots with up to 10 crossings emerge as favoured fold designs for ssRNA, since they are readily constructed as (a)-type all-antiparallel folds.

Zmiany sieci pozamiejskiego publicznego transportu autobusowego w Bieszczadach i Beskidzie Niskim – ujęcie topologiczne

W artykule autor powraca do klasycznych niegdyś w geografii transportu metod grafowych. Wykorzystując podstawowe wskaźniki, takie jak liczba cyklomatyczna μ, wskaźnik α Kansky’ego, wskaźnik γ Kansky’ego i wskaźnik Gns opracowany przez A. Ciechańskiego analizuje zmiany sieci pozamiejskiego autobusowego publicznego transportu zbiorowego na obszarze Beskidu Niskiego i Bieszczad. Testuje też wskaźnik Gns dla bardziej rozbudowanych grafów o skomplikowanej strukturze, w tym również często z bardzo licznymi izolowanymi wierzchołkami. Niestety w przeciwieństwie do prostych i niespójnych sieci transportowych, dla których został on skonstruowany, w przypadku dużych sieci transportowych, zawierających liczne cykle jego czułość wykazuje znacznie gorszy poziom, a otrzymane wyniki są znacznie mniej jednoznaczne niż w przypadku gdy izolowane podgrafy są mniej liczne, za to o bardziej rozbudowanej strukturze. Słowa kluczowe: metody grafowe, wskaźnik Gns, zmiany sieci pozamiejskiego publicznego transportu zbiorowego, Beskid Niski, Bieszczady Changes in the network of the non-urban public bus transport in Bieszczady and Beskid Niski mountains – a topological approach In the article, the author returns to the graph methods which were once classic in the transport geography. Using basic indicators such as the cyclomatic number μ, the α Kansky index, the γ Kansky index and the Gns index developed by A. Ciechański, he analyzes the changes in the network of non-urban public bus transport in the area of the Beskid Niski and the Bieszczady Mountains. He also tests the Gns indicator for more complex graphs with a complicated structure, including often very numerous isolated vertices. Unfortunately, unlike the simple and inconsistent transport networks for which it was created, in the case of large transport networks containing many cycles its sensitivity shows a much worse level and the obtained results are much less unambiguous than in the case when the isolated subgraphs are less numerous, but with the more elaborate structure.

Wiener Complexity versus the Eccentric Complexity

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G).

Main eigenvalues of a graph and its Hamiltonicity

The concept of (κ,τ)-regular vertex set appeared in 2004. It was proved that the existence of many classical combinatorial structures in a graph like perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized by (κ,τ)-regular sets the definition whereof is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)-regular sets is closely related to the properties of the main spectrum of a graph. This paper generalizes the well-known properties of (κ,κ)-regular sets of a graph to arbitrary (κ,τ)-regular sets of graphs with an emphasis on their connection with classical combinatorial structures. We also present a recognition algorithm for the Hamiltonicity of the graph that becomes polynomial in some classes of graphs, for example, in the class of graphs with a fixed cyclomatic number.