arXiv subjects

Chaochao Zhu

Publications and source records attributed to Chaochao Zhu.

5 recordsLinked to original sources

arXiv2026

The Distance Between the Adjacency Spectral Center and the Characteristic Set of a Tree

Let $\mathcal S(T)$ be the adjacency spectral center of a tree $T$, and let $\mathcal C(T)$ be its characteristic set. We determine the largest possible separation $d(T) := \operatorname{dist}_T(\mathcal{S}(T), \mathcal{C}(T))$ among trees of every order $n\ge3$. Writing $Δ_n:=\max_{|V(T)|=n}d(T)$, we prove $Δ_n=0\quad(3\le n\le11),\quad Δ_{12}=1$, and $Δ_n=\left\lfloor\frac{n-11}{2}\right\rfloor \quad(n\ge13)$. The argument rests on a simple opposition between two rooted-tree weights. An endpoint-rooted path minimizes adjacency spectral radius, but maximizes bottleneck Perron value. A one-sided replacement by a path therefore cannot decrease the distance between the two centers. Quantitatively, this gives the sharp estimate $|V(T)|\ge 2d(T)+11 \quad(d(T)\ge2)$. A preliminary six-vertex barrier shows that disjoint center sets require at least twelve vertices, and the four-leaf broom is extremal for every $n\ge12$.

math.CO↗

arXiv2026

Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees

We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let $M_G(x)$ be the main polynomial of a graph $G$, and let $κ_{\mathrm m}(G)$ denote the sum of the multiplicities in $ϕ_G(x)$ of the irreducible factors dividing $M_G(x)$. We prove that every graph $H$ with the same characteristic polynomial and main polynomial as $G$ satisfies $c(H)\leq κ_{\mathrm m}(G)$. Consequently, if $T$ is a tree and $H$ is generalized cospectral with $T$, then $β(H)=c(H)-1\leq κ_{\mathrm m}(T)-1$. We develop the case $κ_{\mathrm m}(T)=2$ in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of $T$ with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star $D(2m,m+1)$ is DGS but not DS whenever $m\geq2$ and neither $m$ nor $2m+2$ is a perfect square. In particular, $D(8t+4,4t+3)$, $t\geq0$, gives an explicit infinite family of such graphs.

math.CO↗

arXiv2026

A Sharp Matching-Number Threshold for Spectral-Walk Determination of Trees

The spectral characterization of graphs is a central problem in spectral graph theory. In this paper we study when a tree is determined, among trees, by its generalized spectrum. We use the equivalent formulation given by the adjacency spectrum together with the total-walk sequence $W_k(G)=\mathbf 1^{\mathsf T}A(G)^k\mathbf 1$. We determine the exact matching-number threshold for this tree-level reconstruction problem. If $T$ and $T'$ are trees with matching number at most 4 and have the same adjacency spectrum and the same total-walk sequence, then $T\cong T'$. Moreover, in this range it is enough to require equality of $W_k$ for $3\le k\le8$. The bound is sharp: for every positive integer $m$ we construct a pair of non-isomorphic trees with matching number 5 having the same adjacency spectrum and identical total-walk sequences. The proof of the positive result is based on a finite-core reduction and an algebraic reconstruction of the possible pendant attachments.

math.CO↗

arXiv2026

Single-exponential bounds for diagonals of D-finite power series

D-finite power series appear ubiquitously in combinatorics, number theory, and mathematical physics. They satisfy systems of linear partial differential equations whose solution spaces are finite-dimensional, which makes them enjoy a lot of nice properties. After attempts by others in the 1980s, Lipshitz was the first to prove that the class they form in the multivariate case is closed under the operation of diagonal. In particular, an earlier work by Gessel had addressed the D-finiteness of the diagonals of multivariate rational power series. In this paper, we give another proof of Gessel's result that fixes a gap in his original proof, while extending it to the full class of D-finite power series. We also provide a single exponential bound on the degree and order of the defining differential equation satisfied by the diagonal of a D-finite power series in terms of the degree and order of the input differential system.

math.CO↗

arXiv2020

On the Existence of Telescopers for Rational Functions in Three Variables

Zeilberger's method of creative telescoping is crucial for the computer-generated proofs of combinatorial and special-function identities. Telescopers are linear differential or ($q$-)recurrence operators computed by algorithms for creative telescoping. For a given class of inputs, when telescopers exist and how to construct telescopers efficiently if they exist are two fundamental problems related to creative telescoping. In this paper, we solve the existence problem of telescopers for rational functions in three variables including 18 cases. We reduce the existence problem from the trivariate case to the bivariate case and some related problems. The existence criteria given in this paper enable us to determine the termination of algorithms for creative telescoping with trivariate rational inputs.

cs.SC↗