For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp exam...
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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...
For a graph $G$ of order $n$, let $\mathcal E(G)$ denote its adjacency energy and let $\alpha(G)$ denote its independence number. A recent theorem of Kumar and Pragada states that $$\mathcal E(G)\ge 2\bigl(n-\alpha(G)\bigr).$$ We determine all graphs attaining equality. More precisely, equality holds if and only if eve...
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$....
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Let $T$ be a tree on $k$ vertices, and write $n=a(k-...
For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the grap...
