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.
Let $\tau(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $\rho_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31--40] obtained a sharp distance spectral radius condition for $\tau(G)\ge k$...
Let $T$ be a tree. Stanley asked whether the chromatic symmetric function $X_T$ determines $T$ up to isomorphism. We approach this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \dots$ and studying the invariant $\Phi_T = (\partial X_T/\partial p_1)|_{p_1 = 0}$. We prove...
The spectral Tur\'an type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recentl...
For a fixed integer $k\ge2$, we establish a sharp adjacency-spectral upper bound for sufficiently large $m$-edge $kK_3$-free graphs. We prove \[ \lambda(G)\le (k-1)+\sqrt{m-k(k-1)}. \] Moreover, equality holds precisely when $(2k-1)\mid m$ and, up to isolated vertices, $G$ is the join of $K_{2k-1}$ with an independent...
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completi...
For a finite simple graph $G$ of positive order, let $s_k(G)$ be the number of its $k$-vertex tree subgraphs. We prove that, among graphs of a fixed order $n$, the ratios $s_k(G)/s_k(K_n)$ form a nonincreasing sequence in $k$. It follows that the complete graph maximizes the mean subtree order: $\mu(G)\leq\mu(K_n)$, wi...
Jun-Gang Chen, Xian'an Jin, Zhuo Li et al.· 0 citations
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