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Shuchao Li

Publications and source records attributed to Shuchao Li.

At least 37 records · Page 2Linked to original sources
arXiv2021

Adjacency eigenvalues of graphs without short odd cycles

It is well known that spectral Turán type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Turán type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Denote by $λ_1$ and $λ_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $λ_1^{2k}+λ_2^{2k}$ of $n$-vertex $\{C_3,C_5,\ldots,C_{2k+1}\}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].

math.CO↗
arXiv2021

HYPER^2: Hyperbolic Poincare Embedding for Hyper-Relational Link Prediction

Link Prediction, addressing the issue of completing KGs with missing facts, has been broadly studied. However, less light is shed on the ubiquitous hyper-relational KGs. Most existing hyper-relational KG embedding models still tear an n-ary fact into smaller tuples, neglecting the indecomposability of some n-ary facts. While other frameworks work for certain arity facts only or ignore the significance of primary triple. In this paper, we represent an n-ary fact as a whole, simultaneously keeping the integrity of n-ary fact and maintaining the vital role that the primary triple plays. In addition, we generalize hyperbolic Poincaré embedding from binary to arbitrary arity data, which has not been studied yet. To tackle the weak expressiveness and high complexity issue, we propose HYPER^2 which is qualified for capturing the interaction between entities within and beyond triple through information aggregation on the tangent space. Extensive experiments demonstrate HYPER^2 achieves superior performance to its translational and deep analogues, improving SOTA by up to 34.5\% with relatively few dimensions. Moreover, we study the side effect of literals and we theoretically and experimentally compare the computational complexity of HYPER^2 against several best performing baselines, HYPER^2 is 49-61 times quicker than its counterparts.

cs.CL↗
arXiv2021

An arithmetic criterion for graphs being determined by their generalized $A_α$-spectrum

Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} introduced the matrix $A_α(G)=αD(G)+(1-α)A(G)$ for $α\in [0, 1].$ The $A_α$-spectrum of a graph $G$ consists of all the eigenvalues (including the multiplicities) of $A_α(G).$ A graph $G$ is said to be determined by the generalized $A_α$-spectrum (or, DGA$_α$S for short) if whenever $H$ is a graph such that $H$ and $G$ share the same $A_α$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, when $α$ is rational, we present a simple arithmetic condition for a graph being DGA$_α$S. More precisely, put $A_{c_α}:={c_α}A_α(G),$ here ${c_α}$ is the smallest positive integer such that $A_{c_α}$ is an integral matrix. Let $\tilde{W}_{α}(G)=\left[{\bf 1},\frac{A_{c_α}{\bf 1}}{c_α},\ldots, \frac{A_{c_α}^{n-1}{\bf 1}}{c_α}\right]$, where ${\bf 1}$ denotes the all-ones vector. We prove that if $\frac{\det \tilde{W}_{α}(G)}{2^{\lfloor\frac{n}{2}\rfloor}}$ is an odd and square-free integer and the rank of $\tilde{W}_{α}(G)$ is full over $\mathbb{F}_p$ for each odd prime divisor $p$ of $c_α$, then $G$ is DGA$_α$S except for even $n$ and odd $c_α\,(\geqslant 3)$. By our obtained results in this paper we may deduce the main results in \cite{0005} and \cite{0002}.

math.CO↗
arXiv2019

On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain

The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$\, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yields the linear hexagonal cylinder chain, written as $R_n$. We obtain explicit formulae for the resistance distance $r_{L_n}(i, j)$ (resp. $r_{R_n}(i,j)$) between any two vertices $i$ and $j$ of $L_n$ (resp. $R_n$). To the best of our knowledge $\{L_n\}_{n=1}^{\infty}$ and $\{R_n\}_{n=1}^{\infty}$ are two nontrivial families with diameter going to $\infty$ for which all resistance distances have been explicitly calculated. We determine the maximum and the minimum resistance distances in $L_n$ (resp. $R_n$). The monotonicity and some asymptotic properties of resistance distances in $L_n$ and $R_n$ are given. As well we give formulae for the Kirchhoff indices of $L_n$ and $R_n$ respectively.

math.CO↗
arXiv2018

The energy of random signed graph

A signed graph $Γ(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $Γ(G)$. The energy of a signed graph $Γ(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(Γ(G))$ of $Γ(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as follows: Let $p, q \ge 0$ be fixed, $0 \le p+q \le 1$. Given a set of $n$ vertices, between each pair of distinct vertices there is either a positive edge with probability $p$ or a negative edge with probability $q$, or else there is no edge with probability $1-(p+ q)$. The edges between different pairs of vertices are chosen independently. In this paper, we obtain an exact estimate of energy for almost all signed graphs. Furthermore, we establish lower and upper bounds to the energy of random multipartite signed graphs.

math.CO↗
arXiv2018

Relation between the H-rank of a mixed graph and the rank of its underlying graph

Given a simple graph $G=(V_G, E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\widetilde{G}$ is obtained from $G$ by orienting some of its edges. Let $H(\widetilde{G})$ denote the Hermitian adjacency matrix of $\widetilde{G}$ and $A(G)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\widetilde{G}$ (resp. $G$), written as $rk(\widetilde{G})$ (resp. $r(G)$), is the rank of $H(\widetilde{G})$ (resp. $A(G)$). Denote by $d(G)$ the dimension of cycle spaces of $G$, that is $d(G) = |E_G|-|V_G|+ω(G)$, where $ω(G),$ denotes the number of connected components of $G$. In this paper, we concentrate on the relation between the $H$-rank of $\widetilde{G}$ and the rank of $G$. We first show that $-2d(G)\leqslant rk(\widetilde{G})-r(G)\leqslant 2d(G)$ for every mixed graph $\widetilde{G}$. Then we characterize all the mixed graphs that attain the above lower (resp. upper) bound. By these obtained results in the current paper, all the main results obtained in \cite{004,1} may be deduced consequently.

math.CO↗
arXiv2017

Some extremal ratios of the distance and subtree problems in binary trees

Among many topological indices of trees the sum of distances $σ(T)$ and the number of subtrees $F(T)$ have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the "local" versions of these invariants, $σ_T(v)$ for the sum of distance from $v$ to all other vertices and $F_T(v)$ for the number of subtrees containing $v$, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of "middle parts" of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as $σ(T)$ and $F(T)$. Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees In this paper this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to $k$-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned.

math.CO↗
arXiv2017

Further results on the expected hitting time, the cover cost and the related invariants of graphs

A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an $n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified, respectively.

math.CO↗
arXiv2017

The quotients between the (revised) Szeged index and Wiener index of graphs

Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.

math.CO↗
arXiv2015

On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on $n$ vertices.

math.CO↗
arXiv2015

On the extremal total reciprocal edge-eccentricity of trees

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $ξ^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we first introduced four edge-grafting transformations to study the mathematical properties of the reciprocal edge-eccentricity of $G$. Using these elegant mathematical properties, we characterize the extremal graphs among $n$-vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. Some sharp bounds on the reciprocal edge-eccentricity of trees are determined.

math.CO↗
arXiv2013

The extremal problems on the inertia of weighted bicyclic graphs

Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively.

math.CO↗
arXiv2013

On the positive and negative inertia of weighted graphs

The number of the positive, negative and zero eigenvalues in the spectrum of the (edge)-weighted graph $G$ are called positive inertia index, negative inertia index and nullity of the weighted graph $G$, and denoted by $i_+(G)$, $i_-(G)$, $i_0(G)$, respectively. In this paper, the positive and negative inertia index of weighted trees, weighted unicyclic graphs and weighted bicyclic graphs are discussed, the methods of calculating them are obtained.

math.CO↗
arXiv2013

On the eccentric distance sum of unicyclic graphs with a given matching number

Let $G = (V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $ξ^d(G)=\sum_{v \in V_G}\,\varepsilon_G(v)D_G(v),$ where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v)=\sum_{u \in V_G}\,d(u,v)$ is the sum of all distances from the vertex $v$. In this paper, we characterize $n$-vertex unicyclic graphs with given matching number having the minimal and second minimal eccentric distance sums, respectively.

math.CO↗
arXiv2013

Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues

The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two $Q$-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main $Q$-eigenvalues, respectively. As a continuance of it, in this paper, all tricyclic graphs with exactly two $Q$-main eigenvalues are characterized.

math.CO↗
arXiv2012

On the spectral moments of trees with a given bipartition

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote $\mathscr{T}_n^{p, q}={T: T$ is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with eigenvalues $λ_1(G), λ_2(G), ..., λ_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}λ_i^k(G)\,(k=0, 1, ..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G)=(S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1\prec_s G_2$ if for some $k\in {1,2,...,n-1}$, $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among $\mathscr{T}_n^{p, q} (4\leqslant p\leqslant q)$ are characterized.

math.CO↗
arXiv2012

On the spectral moment of graphs with given clique number

Let $\mathscr{L}_{n,t}$ be the set of all $n$-vertex connected graphs with clique number $t$\,($2\leq t\leq n)$. For $n$-vertex connected graphs with given clique number, lexicographic ordering by spectral moments ($S$-order) is discussed in this paper. The first $\sum_{i=1}^{\lfloor\frac{n-t-1}{3}\rfloor}(n-t-3i)+1$ graphs with $3\le t\le n-4$, and the last few graphs, in the $S$-order, among $\mathscr{L}_{n,t}$ are characterized. In addition, all graphs in $\mathscr{L}_{n,n}\bigcup\mathscr{L}_{n,n-1}$ have an $S$-order; for the cases $t=n-2$ and $t=n-3$ the first three and the first seven graphs in the set $\mathscr{L}_{n,t}$ are characterized, respectively.

math.CO↗