arXiv subjects
Publications and source records attributed to Qingzhong Ji.
For a graph $G,$ let $α(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture is that $α(G)+α(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this article, we provide a new proof of the Laplacian spread conjecture by means of linear algebra, which is more concise.
Let \( K \) be a number field. We provide quantitative estimates for the size of the Zsigmondy set of an integral ideal sequence generated by iterating a polynomial function \(φ(z) \in K[z]\) at a wandering point \(α\in K.\)
Let $p, k, q$ be positive integers with $p-2 \geqslant k$ and let $K_{p,k}^{q}$ be the generalized pineapple graph which is obtained by joining independent set of $q$ vertices with $k$ vertices of a complete graph $K_{p}.$ In \cite{TSH2}, Haemers et al. constructed graphs which cospectral with $K_{p,1}^{q}.$ In this paper, we determine all graphs which are cospectral with $K_{p,k}^{q}$ by considering the eigenvalues of its adjacency matrix. Moreover, We extend the conclusions of Haemers et al. to a broader context.
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in\mathbb{F}_q[x]$ and $φ(q,p(x))$ be the Euler's totient function of $p(x)$ over $\mathbb{F}_q[x],$ where $\mathbb{F}_q$ is a finite field with $q$ elements. We prove that $φ(q,p(x))|(q^{{\rm deg}(p(x))}-1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3, \; p(x)$ is the product of any $2$ non-associate irreducibes of degree $1;$ or (iii) $q=2,\; p(x)$ is the product of all irreducibles of degree $1,$ all irreducibles of degree $1$ and $2,$ and the product of any $3$ irreducibles one each of degree $1, 2$ and $3$.
Let $X$ be a smooth projective curve over a finite field $\mathbb{F}$ with $q$ elements. For $m\geq 1,$ let $X_m$ be the curve $X$ over the finite field $\mathbb{F}_m$, the $m$-th extension of $\mathbb{F}.$ Let $K_n(m)$ be the $K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$ In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the $l$-primary part $K_n(l^m)\{l\}$ of $K_n(l^m)$. In particular, when $X$ is an elliptic curve $E$ defined over $\mathbb{F},$ our method determines the structure of $K_n(E).$ Our results can be applied to construct an efficient {\bf DL} system in elliptic cryptography.