arXiv subjects

Florian Werner

Publications and source records attributed to Florian Werner.

17 recordsLinked to original sources

arXiv2025

Leaf to leaf path lengths in trees of given degree sequence

For a tree $T$, let $lp(T)$ be the number of different lengths of leaf to leaf paths in $T$. For a degree sequence $s$ of a tree, let ${\rm rad}(s)$ be the minimum radius of a tree with degree sequence $s$. Recently, Di Braccio, Katsamaktsis, Ma, Malekshahian, and Zhao provided a lower bound on $lp(T)$ in terms of the number of leaves and the maximum degree of $T$, answering a related question posed by Narins, Pokrovskiy, and Szabó. Here we show $lp(T)\geq {\rm rad}(s)-\log_2\left({\rm rad}(s)\right)$ for a tree $T$ with no vertex of degree $2$ and degree sequence $s$, and discuss possible improvements and variants.

math.CO↗

arXiv2025

Coloring by Pushing Vertices

Let $G$ be a graph of order $n$, maximum degree at most $Δ$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function $ρ:V(G)\to\mathbb{N}_0$ for which $$σ:V(G)\to\mathbb{N}_0:u\mapsto \left(1+ρ(u)\right)d_G(u)+\sum_{v\in N_G(u)}ρ(v)$$ is a vertex coloring, that is, adjacent vertices receive different values under $σ$. They show the existence of a proper pushing scheme $ρ$ with $\max\{ ρ(u):u\in V(G)\}\leq Δ^2$ and conjecture that this upper bound can be improved to $Δ$. We show their conjecture for cubic graphs and regular bipartite graphs. Furthermore, we show the existence of a proper pushing scheme $ρ$ with $\sum_{u\in V(G)}ρ(u)\leq \left(2Δ^2+Δ\right)n/6$.

math.CO↗

arXiv2024

An Optimization Approach to Degree Deviation and Spectral Radius

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $λ$, minimum degree at least $δ$, and maximum degree at most $Δ$, where $0\leqδ \frac{dn}{\sqrt{2}}. \end{cases}$$ Our results are based on a smoothing technique relating the degree deviation and the largest eigenvalue to low-dimensional non-linear optimization problems.

math.CO↗

arXiv2024

Degree Deviation and Spectral Radius

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $λ$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a conjecture of Nikiforov, we show $λ-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$. For our result, we show that the largest eigenvalue of a graph that arises from a bipartite graph with $m_{A,B}$ edges by adding $m_A$ edges within one of the two partite sets is at most $\sqrt{m_A+m_{A,B}+\sqrt{m_A^2+2m_Am_{A,B}}}$, which is a common generalization of results due to Stanley and Bhattacharya, Friedland, and Peled.

math.CO↗

arXiv2024

Complexity of Deciding the Equality of Matching Numbers

A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.

cs.DM↗

arXiv2024

Induced Subforests and Superforests

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of forests, we study maximum common induced subforests and minimum common induced superforests. We show that finding a maximum subforest is NP-hard already for two subdivided stars while finding a minimum superforest is tractable for two trees but NP-hard for three trees. For a given set of $k$ trees, we present an efficient greedy $\left(\frac{k}{2}-\frac{1}{2}+\frac{1}{k}\right)$-approximation algorithm for the minimum superforest problem. Finally, we present a polynomial time approximation scheme for the maximum subforest problem for any given set of forests.

cs.DS↗

arXiv2024

Largest common subgraph of two forests

A common subgraph of two graphs $G_1$ and $G_2$ is a graph that is isomorphic to subgraphs of $G_1$ and $G_2$. In the largest common subgraph problem the task is to determine a common subgraph for two given graphs $G_1$ and $G_2$ that is of maximum possible size ${\rm lcs}(G_1,G_2)$. This natural problem generalizes the well-studied graph isomorphism problem, has many applications, and remains NP-hard even restricted to unions of paths. We present a simple $4$-approximation algorithm for forests, and, for every fixed $ε\in (0,1)$, we show that, for two given forests $F_1$ and $F_2$ of order at most $n$, one can determine in polynomial time a common subgraph $F$ of $F_1$ and $F_2$ with at least ${\rm lcs}(F_1,F_2)-εn$ edges. Restricted to instances with ${\rm lcs}(F_1,F_2)\geq cn$ for some fixed positive $c$, this yields a polynomial time approximation scheme. Our approach relies on the approximation of the given forests by structurally simpler forests that are composed of copies of only $O(\log (n))$ different starlike rooted trees and iterative quantizations of the options for the solutions.

cs.DS↗

arXiv2023

Mostar index and bounded maximum degree

Došlić et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and maximum degree at most $Δ$, we show $Mo(G)\leq \fracΔ{2}n^2-(1-o(1))c_Δn\log(\log(n)),$ where $c_Δ>0$ only depends on $Δ$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $Δ$ at least $3$, we show the existence of a $Δ$-regular graph of order $n$ at least $n_0$ with $Mo(G)\geq \fracΔ{2}n^2-c'_Δn\log(n),$ where $c'_Δ>0$ only depends on $Δ$.

math.CO↗

arXiv2023

Irregularity of Graphs respecting Degree Bounds

Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $Δ$, and $d=\left\lfloor \frac{Δm}{Δn-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}Δ\left(Δ^2-(2d+1)Δ-d^2-d\right)m.$$

math.CO↗

arXiv2022

Bounding the Mostar index

Došlić et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. They conjectured that $Mo(G)\leq 0.\overline{148}n^3$ for every graph $G$ of order $n$. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter $Mo^\star(G)=\sum\limits_{uv\in E(G)}\big(n-\min\{ d_G(u),d_G(v)\}\big)$. For a graph $G$ of order $n$, they show that $Mo^\star(G)\leq \frac{5}{24}(1+o(1))n^3$. We improve this bound to $Mo^\star(G)\leq \left(\frac{2}{\sqrt{3}}-1\right)n^3$, which is best possible up to terms of lower order. Furthermore, we show that $Mo^\star(G)\leq \left(2\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)-2\left(\fracΔ{n}\right)\sqrt{\left(\fracΔ{n}\right)^2+\left(\fracΔ{n}\right)}\right)n^3$ provided that $G$ has maximum degree $Δ$.

math.CO↗

arXiv2022

Maximizing the Mostar index for bipartite graphs and split graphs

Došlić et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.

math.CO↗

arXiv2021

Near surface defects: Cause of deficit between internal and external open-circuit voltage in solar cells

The presence of interface recombination in a complex multilayered thin-film solar structure causes a disparity between the internal open-circuit voltage (VOC,in), measured by photoluminescence, and the external open-circuit voltage (VOC,ex) i.e. an additional VOC deficit. Higher VOC,ex value aim require a comprehensive understanding of connection between VOC deficit and interface recombination. Here, a deep near-surface defect model at the absorber/buffer interface is developed for copper indium di-selenide solar cells grown under Cu excess conditions to explain the disparity between VOC,in and VOC,ex.. The model is based on experimental analysis of admittance spectroscopy and deep-level transient spectroscopy, which show the signature of deep acceptor defect. Further, temperature-dependent current-voltage measurements confirm the presence of near surface defects as the cause of interface recombination. The numerical simulations show strong decrease in the local VOC,in near the absorber/buffer interface leading to a VOC deficit in the device. This loss mechanism leads to interface recombination without a reduced interface bandgap or Fermi level pinning. Further, these findings demonstrate that the VOC,in measurements alone can be inconclusive and might conceal the information on interface recombination pathways, establishing the need for complementary techniques like temperature dependent current voltage measurements to identify the cause of interface recombination in the devices.

cond-mat.mtrl-sci↗

arXiv2021

Passivating Surface Defects and Reducing Interface Recombination in CuInS2 Solar Cells by a Facile Solution Treatment

Interface recombination at the absorber buffer interface impedes the efficiency of a solar cell with an otherwise excellent absorber. The internal voltage or the quasi-Fermi level splitting (qFLs) measures the quality of the absorber. Interface recombination reduces the open circuit voltage (VOC) with respect to the qFLs. The present work explores a facile sulfur-based post-deposition treatment (S-PDT) to passivate the interface of CuInS2 thin films grown under Cu-rich conditions, which show excellent qFLs values, but much lower VOCs. The CuInS2 absorbers are treated in three different S-containing solutions at 80 oC. Absolute calibrated photoluminescence and current-voltage measurements demonstrate a reduction of the deficit between qFLs and VOC in the best S-PDT device by almost one third compared to the untreated device. Analysis of temperature dependence of the open-circuit voltage shows increased activation energy for the dominant recombination path, indicating less interface recombination. In addition, capacitance transient measurements reveal the presence of slow metastable defects in the untreated solar cell. The slow response is considerably reduced by the S-PDT, suggesting passivation of these slow metastable defects. The results demonstrate the effectiveness of solution based S-treatment in passivating defects, presenting a promising strategy to explore and reduce defect states near the interface of chalcogenide semiconductors.

cond-mat.mtrl-sci↗

arXiv2018

Brownian Motions on Metric Graphs with Non-Local Boundary Conditions II: Construction

A pathwise construction of discontinuous Brownian motions on metric graphs is given for every possible set of non-local Feller-Wentzell boundary conditions. This construction is achieved by locally decomposing the metric graphs into star graphs, establishing local solutions on these partial graphs, pasting the solutions together, introducing non-local jumps, and verifying the generator of the resulting process.

math.PR↗

arXiv2018

Brownian Motions on Metric Graphs with Non-Local Boundary Conditions I: Characterization

A classification for Brownian motions on metric graphs, that is, right continuous strong Markov processes which behave like a one-dimensional Brownian motion on the edges and feature effects like Walsh skewness, stickiness and jumps at the vertices, is obtained. The Feller property of these processes is proved, and the boundary conditions of their generators are identified as non-local Feller-Wentzell boundary conditions. By using a technique of successive revivals, a complete description of the generator is achieved for Brownian motions on star graphs.

math.PR↗

arXiv2018

Brownian Motions on Star Graphs with Non-Local Boundary Conditions

Brownian motions on star graphs in the sense of Itô-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for infinite jump measures, the pathwise solution of Itô-McKean for the half line is analyzed and extended to the star graph. These processes can then be used as main building blocks for Brownian motions on general metric graphs with non-local boundary conditions.

math.PR↗

arXiv2018

Concatenation and Pasting of Right Processes

A universal method for the concatenation of a sequence of Markov right processes is established. It is then applied to the continued pasting of two Markov right processes, which can be used for pathwise constructions of locally defined processes like Brownian motions on compact intervals.

math.PR↗