Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfying $S_k(G)<2k$. As a consequence, we prove that the path $P_n$ is the unique minimizer of $S_k(G)$ among all connected graphs of order $n$. \vspace*{2mm} We further investigate the first Hoffman-type range \[ 2k\leq S_k(G)<2k+\sqrt{2+\sqrt5}-2. \] We completely characterize the non-tree graphs in this range and reduce the tree case to several explicit families. The proofs combine Ky Fan's variational principle, spectral estimates from vertex-disjoint subgraphs, structural results for graphs with small spectral radius, and long-path arguments for bounded-degree graphs.
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
Let $k\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\ge 2k-1$. More precisely, we prove that every such gra...
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 $\tau(G)\ge k$...
Extending the notions of Kneser graphs and Schrijver graphs, we define the $G$-stable Kneser graph $KG(G,k)$ (where $k\geq 2$ and $G$ is a graph on $n\geq 2k$ vertices) as the graph whose vertices are the $k$-element independent sets in $G$ and whose edges join pairs of disjoint independent sets. The well-known result...
C. Arana, P. Haxell, Tomáš Kaiser et al.· 0 citations
Let $S_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of a connected graph $G$ of order $n$ and size $m$. Write $\mathrm{PA}_{n,\omega}$ for the graph obtained from an $\omega$-vertex clique by attaching $n-\omega$ pendant vertices to one of its vertices, and set \[ M_{n,k}:=\binom{k+1}{2}+n-k-1, \] the...
Let $\lambda_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $\varphi_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $\varphi_k(G)\le C\...
Yun-Peng Li· 0 citations
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