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Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems

Francis Clarke, John Hunton, Nigel Ray (2001)

Annales de l’institut Fourier

We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring E * with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where E * is free of additive torsion, in which context the central issues are number- theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions...

Extraresolvability of balleans

Igor V. Protasov (2007)

Commentationes Mathematicae Universitatis Carolinae

A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpart of a uniform topological space. We introduce and study a new cardinal invariant of a ballean, the extraresolvability, which is an asymptotic reflection of the corresponding invariant of a topological space.

Extremal bipartite graphs with a unique k-factor

Arne Hoffmann, Elżbieta Sidorowicz, Lutz Volkmann (2006)

Discussiones Mathematicae Graph Theory

Given integers p > k > 0, we consider the following problem of extremal graph theory: How many edges can a bipartite graph of order 2p have, if it contains a unique k-factor? We show that a labeling of the vertices in each part exists, such that at each vertex the indices of its neighbours in the factor are either all greater or all smaller than those of its neighbours in the graph without the factor. This enables us to prove that every bipartite graph with a unique k-factor and maximal size...

Extremal inverse eigenvalue problem for matrices described by a connected unicyclic graph

Bijoya Bardhan, Mausumi Sen, Debashish Sharma (2024)

Applications of Mathematics

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic...

Extremal Matching Energy of Complements of Trees

Tingzeng Wu, Weigen Yan, Heping Zhang (2016)

Discussiones Mathematicae Graph Theory

Gutman and Wagner proposed the concept of the matching energy which is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. And they pointed out that the chemical applications of matching energy go back to the 1970s. Let T be a tree with n vertices. In this paper, we characterize the trees whose complements have the maximal, second-maximal and minimal matching energy. Furthermore, we determine the trees with edge-independence number p whose complements have...

Extremal problems for forbidden pairs that imply hamiltonicity

Ralph Faudree, András Gyárfás (1999)

Discussiones Mathematicae Graph Theory

Let C denote the claw K 1 , 3 , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and Z i the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does...

Extremal Unicyclic Graphs With Minimal Distance Spectral Radius

Hongyan Lu, Jing Luo, Zhongxun Zhu (2014)

Discussiones Mathematicae Graph Theory

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) Cn.

Extreme geodesic graphs

Gary Chartrand, Ping Zhang (2002)

Czechoslovak Mathematical Journal

For two vertices u and v of a graph G , the closed interval I [ u , v ] consists of u , v , and all vertices lying in some u -- v geodesic of G , while for S ⊆ V ( G ) , the set I [ S ] is the union of all sets I [ u , v ] for u , v ∈ S . A set S of vertices of G for which I [ S ] = V ( G ) is a geodetic set for G , and the minimum cardinality of a geodetic set is the geodetic number g ( G ) . A vertex v in G is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in G is its extreme order e x ( G ) . A graph G is an extreme geodesic...

Extremely primitive groups and linear spaces

Haiyan Guan, Shenglin Zhou (2016)

Czechoslovak Mathematical Journal

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let 𝒮 be a nontrivial finite regular linear space and G ≤ Aut ( 𝒮 ) . Suppose that G is extremely primitive on points and let rank ( G ) be the rank of G on points. We prove that rank ( G ) ≥ 4 with few exceptions. Moreover, we show that Soc ( G ) is neither a sporadic group nor an alternating group, and G = PSL ( 2 , q ) with q + 1 a Fermat prime if Soc ( G ) is a finite classical simple group.

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