arXiv subjects

Yiyang Li

Publications and source records attributed to Yiyang Li.

63 records · Page 4Linked to original sources

arXiv2013

The asymptotic number of different rooted trees of a tree

Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(μ_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $μ_r$ is a constant. As an application, we show that the number of any given pattern in $\mathcal{T}_n$ is also asymptotically normally distributed with mean $\sim μ_M n$ and variance $\sim σ_M n$, where $μ_M, σ_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.

math.CO↗

arXiv2010

The asymptotic number of occurrences of a subtree in trees with bounded maximum degree and an application to the Estrada index

Let $\mathcal {T}^Δ_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $Δ$. Suppose that every tree in $\mathcal {T}^Δ_n$ is equally likely. For any given subtree $H$, we show that the number of occurrences of $H$ in trees of $\mathcal {T}^Δ_n$ is with mean $(μ_H+o(1))n$ and variance $(σ_H+o(1))n$, where $μ_H$, $σ_H$ are some constants. As an application, we estimate the value of the Estrada index $EE$ for almost all trees in $\mathcal {T}^Δ_n$, and give an explanation in theory to the approximate linear correlation between $EE$ and the first Zagreb index obtained by quantitative analysis.

math.CO↗

arXiv2010

The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degree

Let $\mathcal {T}^Δ_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $Δ$. Suppose that every tree in $\mathcal {T}^Δ_n$ is equally likely. We show that the number of vertices of degree $j$ in $\mathcal {T}^Δ_n$ is asymptotically normal with mean $(μ_j+o(1))n$ and variance $(σ_j+o(1))n$, where $μ_j$, $σ_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in $\mathcal {T}^Δ_n$. Moreover, we obtain that the number of edges of type $(i,j)$ in $\mathcal {T}^Δ_n$ also has mean $(μ_{ij}+o(1))n$ and variance $(σ_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and $μ_{ij}$, $σ_{ij}$ are some constants. Then, we give estimate to the value of the general Randić index for almost all trees in $\mathcal {T}^Δ_n$.

math.CO↗

arXiv2010

The asymptotic value of Randic index for trees

Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with $n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of $\mathcal{T}_n$ is equally likely, we show that the limiting distribution of the number of occurrences of the double-star $(i,j)$ in $\mathcal{T}_n$ is normal. Based on this result, we obtain the asymptotic value of Randić index for trees. Fajtlowicz conjectured that for any connected graph the Randić index is at least the average distance. Using this asymptotic value, we show that this conjecture is true not only for almost all connected graphs but also for almost all trees.

math.CO↗

arXiv2009

The Laplacian energy of random graphs

Gutman {\it et al.} introduced the concepts of energy $\En(G)$ and Laplacian energy $\EnL(G)$ for a simple graph $G$, and furthermore, they proposed a conjecture that for every graph $G$, $\En(G)$ is not more than $\EnL(G)$. Unfortunately, the conjecture turns out to be incorrect since Liu {\it et al.} and Stevanović {\it et al.} constructed counterexamples. However, So {\it et al.} verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs.

math.CO↗

arXiv2009

The energy of random graphs

In 1970s, Gutman introduced the concept of the energy $\En(G)$ for a simple graph $G$, which is defined as the sum of the absolute values of the eigenvalues of $G$. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner's semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.

math.CO↗

arXiv2009

Note on the energy of regular graphs

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known inequality is that $\mathcal{E}(G)\leq λ_1+\sqrt{(n-1)(2m-λ_1)}$, where $λ_1$ is the spectral radius. If $G$ is $k$-regular, we have $\mathcal{E}(G)\leq k+\sqrt{k(n-1)(n-k)}$. Denote $\mathcal{E}_0=k+\sqrt{k(n-1)(n-k)}$. Balakrishnan [{\it Linear Algebra Appl.} {\bf 387} (2004) 287--295] proved that for each $ε>0$, there exist infinitely many $n$ for each of which there exists a $k$-regular graph $G$ of order $n$ with $k< n-1$ and $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<ε$, and proposed an open problem that, given a positive integer $n\geq 3$, and $ε>0$, does there exist a $k$-regular graph $G$ of order $n$ such that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-ε$. In this paper, we show that for each $ε>0$, there exist infinitely many such $n$ that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}>1-ε$. Moreover, we construct another class of simpler graphs which also supports the first assertion that $\frac{\mathcal{E}(G)}{\mathcal{E}_0}<ε$.

math.CO↗

arXiv2009

The limiting spectral distribution of the generalized Wigner matrix

The properties of eigenvalues of large dimensional random matrices have received considerable attention. One important achievement is the existence and identification of the limiting spectral distribution of the empirical spectral distribution of eigenvalues of Wigner matrix. In the present paper, we explore the limiting spectral distribution for more general random matrices, and, furthermore, give an application to the energy of general random graphs, which generalizes the result of Nikiforov.

math.CO↗

arXiv2007

Filtrations in Modular Representations of Reductive Lie Algebras

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with $p$-character $χ$ of standard Levi-form associated with an index subset $I$ of simple roots. With aid of support variety theory we prove a theorem that a $U_χ(\ggg)$-module is projective if and only if it is a strong "tilting" module, i.e. admitting both $\cz_Q$- and $\cz^{w^I}_Q$-filtrations (to see Theorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{AK} for $G_1T$-modules, we construct so-called Andersen-Kaneda filtrations associated with each projective $\ggg$-module of $p$-character $χ$, and finally obtain sum formulas from those filtrations.

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