arXiv subjects

Yujun Yang

Publications and source records attributed to Yujun Yang.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorial explanation of the weighted Kirchhoff index of graphs

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $\omega:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $\omega(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to $2$-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.

math.CO↗

arXiv2026

Resistance Curvature: Recognition, Polyhedral Structure, and Graph Products

Resistance curvature, introduced by Devriendt and Lambiotte, is a novel discrete curvature notion defined through effective resistance. A graph is called resistance nonnegative if there exists a choice of positive edge weights for which the resistance curvature is nonnegative at every vertex. This property has a notable combinatorial interpretation in terms of random spanning trees: a graph is resistance nonnegative if and only if it admits a distribution on its spanning trees under which every vertex has expected degree at most two. The resistance nonnegativity can also be characterized by the tree double matching polytope. These characterizations reveal strong connections among resistance curvature, effective resistance, spanning tree distributions, matching theory, and polyhedral combinatorics. Based on the sign of the curvature, Devriendt introduced the classes of resistance nonnegative (RN), resistance positive (RP), and strictly resistance nonnegative (SRN) graphs, and posed several questions concerning their recognition, polyhedral structure, and structural properties. In this paper, we first answer Devriendt's question on the computational complexity of recognizing RN, RP, and SRN graphs by proving that all three classes can be recognized in polynomial time. We then address his question concerning the tree double matching polytope $\Theta(G)$. Further, we characterize the vertices of $\Theta(G)$ in terms of full-rank systems of tight constraints. Whenever $\Theta(G)\neq\emptyset$, we also determine the least positive integer $k_G$ such that $k_G\Theta(G)$ is a lattice polytope. Finally, for every finite Cartesian product of paths, we explicitly construct an average point satisfying the condition for resistance nonnegativity, thereby obtaining that such graphs are RN. We further characterize the classes of such Cartesian product graphs that are RP or SRN.

math.CO↗

arXiv2026

On the Lei--Bai conjecture on $5$-regular Lin--Lu--Yau Ricci-flat graphs

We study the Ricci curvature introduced by Lin, Lu, and Yau. A graph is called Ricci-flat if every edge has curvature zero. Lei and Bai classified $5$-regular symmetric Ricci-flat graphs by proving that every such graph is isomorphic to a particular $72$-vertex graph $\RF$, and conjectured that every $5$-regular Ricci-flat graph is either isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. In this paper, we disprove this conjecture by constructing an infinite family of connected $5$-regular Ricci-flat graphs, none of which is isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. This shows that the conjectured extension of the classification from the symmetric setting to general $5$-regular Ricci-flat graphs fails and that the class of such graphs is substantially richer than previously conjectured. To establish these results, we use an optimal-assignment formulation of Lin--Lu--Yau curvature to verify the Ricci-flatness of the constructed graphs.

math.CO↗

arXiv2026

On two conjectures concerning Kemeny's constant of graphs

Kemeny's constant for a connected graph $G$, denoted by $\mathcal{K}(G)$, is the expected time for a random walk to reach a randomly chosen vertex $u$, regardless of the choice of the initial vertex. Recently, Kim et al. (2026) proposed two conjectures on Kemeny's constant. The first conjecture asserts that if $G$ is a connected graph of order $n$ and diameter 2, then $\mathcal{K}(G) = O(n)$. The second conjecture asserts that if $G$ be a graph of order $n$, then $\min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} = O(n)$, and if both $G$ and $\overline{G}$ are connected, then $\mathcal{K}(G)\mathcal{K}(\overline{G}) = O(n^4)$, where $\overline{G}$ denotes the complement of $G$. In this paper, we confirm both conjectures. For the first conjecture, we prove that if $G$ is a connected graph of order $n$ and diameter 2, then \[ \mathcal{K}(G) \leq (3 + \sqrt{5})(n - 1). \] For the second conjecture, we prove that for any $n$-vertex graph $G$, \[ \min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} \leq (8 + 2\sqrt{5})n - (10 + 2\sqrt{5}). \] Moreover, if both $G$ and $\overline{G}$ are connected, then \[ \mathcal{K}(G)\mathcal{K}(\overline{G}) \leq \frac{3 + \sqrt{5}}{2}n^4. \] Our proof relies on effective estimates on resistance distances and spectral gaps of graphs.

math.CO↗

arXiv2026

The size of $k$-th order generalized Fibonacci cubes

Let $k\geq2$. Then the $k$-th order Fibonacci cube $\Gamma^{(k)}_{n}$ is the subgraph of the hypercube $Q_{n}$ induced by vertices without $k$ consecutive $1$s. The case $k=2$ corresponds to the classic Fibonacci cube $\Gamma_{n}$. There are three kinds of calculation formulas of the size of $\Gamma_{n}$: the iteration form $|E(\Gamma_{n})|=|E(\Gamma_{n-1})|+|E(\Gamma_{n-2})|+F_{n}$ (Hsu, 1993), %iteration form the convolution form $|E(\Gamma_{n})|=\mathop{\sum}\limits_{i=1}^{n}F_{i}F_{n-i+1}$ (Klav\v{z}ar, 2005) %convolution form and the linear form $|E(\Gamma_{n})|=\frac{nF_{n+1}+2(n+1)F_{n}}{5}$ (Munarini et al., 2001). %linear form Belbachir and Ould-Mohamed (2020) studied the iteration and convolution formulas of the size of $\Gamma^{(3)}_{n}$. Very recently, Mollard (2025) deduced the iteration formula of the size of $\Gamma^{(k)}_{n}$ for $k\geq2$. In this paper, we give the the formulas of convolution and linear forms of $|E(\Gamma^{(k)}_{n})|$ for all $k\geq2$. Specifically, we obtain the formula of $|E(\Gamma^{(k)}_{n})|$ in terms of convolved $k$-th order Fibonacci numbers and the formula of $|E(\Gamma^{(k)}_{n})|$ of linear expression of $k$ consecutive $k$-th order Fibonacci numbers.

math.CO↗

arXiv2025

A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes

A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching.

math.CO↗

arXiv2025

Odd clique minors and chromatic bounds of {3$K_1$, paraglider}-free graphs

A paraglider, house, 4-wheel, is the graph that consists of a cycle $C_4$ plus an additional vertex adjacent to three vertices, two adjacent vertices, all the vertices of the $C_4$, respectively. For a graph $G$, let $\chi(G)$, $\omega(G)$ denote the chromatic number, the clique number of $G$, respectively. Gerards and Seymour from 1995 conjectured that every graph $G$ has an odd $K_{\chi(G)}$ minor. In this paper, based on the description of graph structure, it is shown that every graph $G$ with independence number two satisfies the conjecture if one of the following is true: $\chi(G) \leq 2\omega(G)$ when $n $ is even, $\chi(G) \leq 9\omega(G)/5$ when $n$ is odd, $G$ is a quasi-line graph, $G$ is $H$-free for some induced subgraph $H$ of paraglider, house or $W_4$. Moreover, we derive an optimal linear $\chi$-binding function for {3$K_1$, paraglider}-free graph $G$ that $\chi(G)\leq \max\{\omega(G)+3, 2\omega(G)-2\}$, which improves the previous result, $\chi(G)\leq 2\omega(G)$, due to Choudum, Karthick and Shalu in 2008.

math.CO↗

arXiv2025

On the minimum constant resistance curvature conjecture of graphs

Let $G$ be a connected graph with $n$ vertices. The resistance distance $\Omega_{G}(i,j)$ between any two vertices $i$ and $j$ of $G$ is defined as the effective resistance between them in the electrical network constructed from $G$ by replacing each edge with a unit resistor. The resistance matrix of $G$, denoted by $R_G$, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $\Omega_{G}(i,j)$. The resistance curvature $\kappa_i$ in the vertex $i$ is defined as the $i$-th component of the vector $(R_G)^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes the all-one vector. If all the curvatures in the vertices of $G$ are equal, then we say that $G$ has constant resistance curvature. Recently, Devriendt, Ottolini and Steinerberger \cite{kde} conjectured that the cycle $C_n$ is extremal in the sense that its curvature is minimum among graphs with constant resistance curvature. In this paper, we confirm the conjecture. As a byproduct, we also solve an open problem proposed by Xu, Liu, Yang and Das \cite{kxu} in 2016. Our proof mainly relies on the characterization of maximum value of the sum of resistance distances from a given vertex to all the other vertices in 2-connected graphs.

math.CO↗

arXiv2024

Associated Mersenne graphs

In this paper, a new sub-family of Hypercubes called the \textit{associated Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated Mersenne graphs is motivated from the Fibonacci-run graphs ({\"O}. E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c}, 2021) by extending run-constrained strings to circularly-run-constrained strings. The name of this new family of graphs is identified with the interesting fact that $|V(\mathcal{M}_{n})|$ is equal to the $n$-th associated Mersenne number. Various interesting structural and enumerative properties of associated Mersenne graphs are investigated, including the analogue of the fundamental recursion, number of vertices and edges, radius, diameter, center, periphery and medianicity. Some future research directions and open problems concerning associated Mersenne graphs are also proposed.

math.CO↗

arXiv2024

Solution to a conjecture on resistance distances of block tower graphs

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. The resistance diameter of $G$, denoted by $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture.

math.CO↗

arXiv2024

On a conjecture of E\v{g}ecio\v{g}lu and Ir\v{s}i\v{c}

In 2021, {\"O}. E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c} introduced the concept of Fibonacci-run graph $\mathcal{R}_{n}$ as an induced subgraph of Hypercube. They conjectured that the diameter of $\mathcal{R}_{n}$ is given by $n-\lfloor(1+\frac{n}{2})^{\frac{1}{2}}-\frac{3}{4}\rfloor$. In this paper, we introduce the novel concept of distance-barriers between vertices in $\mathcal{R}_{n}$ and provide an elegant method to give lower bound for the diameter of $\mathcal{R}_{n}$ via distance-barriers. By constructing different types of distance-barriers, we show that the conjecture does not hold for all $n\geq 230$ and some of $n$ between $91$ and $229$. Furthermore, lower bounds for the diameter of some Fibonacci-run graphs are obtained, which turn out to be better than the result given in the conjecture.

math.CO↗

arXiv2023

Imagining density distribution of molecular orbitals in IR+XUV co-rotating circular laser fields by frequency-domain theory

We have investigated the angle-resolved ATI spectrum of oriented molecules in the IR+XUV co-rotating circular laser fields. According to the different roles of IR and XUV laser in the ionization process, we purposefully adjust the photon energy of XUV and the intensity of IR laser to make the ionization spectrum of the molecule distributed in a suitable momentum region. Moreover, under the same laser conditions, the background fringes in the ionization spectrum of the molecule can be removed by using the ionization spectrum of the atom with the same ionization energy as the molecule, so that the molecular orbital density distribution in the suitable momentum region can be obtained. That is, for any unknown molecule, as long as the ionization energy of the molecule can be measured, the density distribution of the molecular orbital can be imaged in a definite momentum region by adjusting the laser field conditions, which may shed light on the experimental detection of molecular orbitals.

physics.atom-ph↗

arXiv2022

On spanning tree edge denpendences of graphs

Let $τ(G)$ and $τ_G(e)$ be the number of spanning trees of a connected graph $G$ and the number of spanning trees of $G$ containing edge $e$. The ratio $d_{G}(e)=τ_{G}(e)/τ(G)$ is called the spanning tree edge density of $e$, or simply density of $e$. The maximum density $\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree edge dependence of $G$, or simply dependence of $G$. Given a rational number $p/q\in (0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that $d_{G}(e)=p/q$, then we say the density $p/q$ is constructible. More specially, if there exists a graph $G$ such that $\mbox{dep}(G)=p/q$, then we say the dependence $p/q$ is constructible. In 2002, Ferrara, Gould, and Suffel raised the open problem of which rational densities and dependences are constructible. In 2016, Kahl provided constructions that show all rational densities and dependences are constructible. Moreover, He showed that all rational densities are constructible even if $G$ is restricted to bipartite graphs or planar graphs. He thus conjectured that all rational dependences are also constructible even if $G$ is restricted to bipartite graphs (Conjecture 1), or planar graphs (Conjecture 2). In this paper, by combinatorial and electric network approach, firstly, we show that all rational dependences are constructible via bipartite graphs, which confirms the first conjecture of Kahl. Secondly, we show that all rational dependences are constructible for planar multigraphs, which confirms Kahl's second conjecture for planar multigraphs. However, for (simple) planar graphs, we disprove the second conjecture of Kahl by showing that the dependence of any planar graph is larger than $\frac{1}{3}$. On the other hand, we construct a family of planar graphs that show all rational dependences $p/q>\frac{1}{2}$ are constructible via planar graphs.

math.CO↗

arXiv2020

On the Fibonacci $(p,r)$-cubes

In this paper, first it is shown that the "FSibonacci $(p,r)$-cube"(denoted as $IΓ_{n}^{(p,r)}$) studied in many papers, such as \cite{OZY}, \cite{K1}, \cite{OZ}, \cite{KR} and \cite{JZ}, is a new topological structure different from the original one (denoted as $OΓ_{n}^{(p,r)}$) presented by Egiazarian and Astola $\cite{EA}$. Then some topological properties of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$ are studied, including the recursive structure of them, the cubes $OΓ_{n}^{(p,r)}$ which are partial cubes and median graphs, some distance invariants of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$, and the maximum and minimum degree of these two types of cubes. Finally, several problems and conjectures on $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$ are listed

math.CO↗

arXiv2019

Cooper minimum of high-order harmonic spectra from MgO crystal in an ultrashort laser pulse

Cooper minimum structure of high-order harmonic spectra from atoms or molecules has been extensively studied. In this paper, we demonstrate that the crystal harmonic spectra from an ultrashort mid-infrared laser pulse also exhibit the Cooper minimum characteristic. Based on the accurate band dispersion and k-dependent transition dipole moment (TDM) from the first-principle calculations, it can be found that the harmonic spectra from MgO crystal along Γ-X direction present a dip structure in the plateau, which is originated from the valley of TDM by examining the distribution of the harmonic intensity at the k-space. The Cooper minimum feature in crystal HHG will pave a new way to retrieve the band information of solid materials by using HHG from the ultrashort mid-infrared laser pulse.

physics.atm-clus↗

arXiv2019

Understanding two-photon double ionization of helium from the perspective of the characteristic time of dynamic transitions

By using the B-spline numerical method, we investigate a two-photon double-ionization (TPDI) process of helium in a high-frequency laser field with its frequency ranging from 1.6~a.u. to 3.0~a.u. and the pulse duration ranging from 75 to 160~attoseconds. We found that there exists a characteristic time $t_{c}$ for a TPDI process, such that the pattern of energy distribution of two ionized electrons presents a peak or two, depending respectively on whether the pulse duration is shorter or longer than $t_{c}$. Especially, as the pulse duration is larger than $t_c$, the TPDI spectrum shows a double-peak structure which is attributed to the fact that most of the electron-electron Coulomb interaction energy is acquired by single electron during their oscillation around the nucleus before the two electrons leave. Additionally, if the photon energy is less than the ionization energy of He$^{+}$, $t_{c}$ is not a fixed value, and it increases as the photon energy decreases; while if the energy of a photon is greater than the ionization energy of He$^{+}$, $t_{c}$ is fixed at about 105 attoseconds. We further found that, for a helium-like ion in its ground state, the characteristic time for the case of the photon energy larger than the ionization energy of the second electron has a key relation with the Coulomb interaction energy $\overline{V}_{12}$ between the two electrons, which can be expressed as $t_{c}\overline{V}_{12}=4.192$, a type of quantum mechanical uncertainty relation between time and energy. In addition, this relation can be attributed to the existence of a minimal evolution time from the ground state to a double ionization state with two electrons carrying different energies. These results may shed light on deeper understanding of many-electron quantum dynamical processes.

physics.atom-ph↗

arXiv2017

Nonsequential double ionization of helium in IR+XUV two-color laser fields II: Collision-excitation ionization process

The collision-ionization mechanism of nonsequential double ionization (NSDI) process in IR+XUV two-color laser fields [\PRA \textbf{93}, 043417 (2016)] has been investigated by us recently. Here we extend this work to study the collision-excitation-ionization (CEI) mechanism of NSDI processes in the two-color laser fields with different laser conditions. It is found that the CEI mechanism makes a dominant contribution to the NSDI as the XUV photon energy is smaller than the ionization threshold of the He$^+$ ion, and the momentum spectrum shows complex interference patterns and symmetrical structures. By channel analysis, we find that, as the energy carried by the recollision electron is not enough to excite the bound electron, the bound electron will absorb XUV photons during their collision, as a result, both forward and backward collisions make a comparable contributions to the NSDI processes. However, it is found that, as the energy carried by the recollision electron is large enough to excite the bound electron, the bound electron does not absorb any XUV photon and it is excited only by sharing the energy carried by the recollsion electron, hence the forward collision plays a dominant role on the NSDI processes. Moreover, we find that the interference patterns of the NSDI spectra can be reconstructed by the spectra of two above-threshold ionization (ATI) processes, which may be used to analyze the structure of the two separate ATI spectra by NSDI processes.

physics.atom-ph↗

arXiv2016

Inverses of Bipartite Graphs

Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].

math.CO↗