The first analytical proof of optimal quantum search on the simplex of complete graphs is given, proving that it occurs when the weight of the bridges equals $M$ and that the optimal runtime is achieved more broadly whenever the weight is at least $\sqrt{M}$.
Abstract
The simplex of complete graphs, also known as the first-order truncated simplex lattice, is a network of $M+1$ identical complete graphs, each with $M$ vertices, such that each clique contains an edge or bridge to every other clique. It contains $N = M(M+1)$ vertices, and previous asymptotic results using a continuous-time quantum walk to search this graph for a single marked vertex have either numerically demonstrated an optimal runtime of $O(\sqrt{N})$, or analytically proved a deterministic success probability of 1, but not both, even when the bridges are weighted. In this paper, we give the first analytical proof of optimal quantum search on this graph, proving that it occurs when the weight of the bridges equals $M$. In addition, we numerically show that the optimal runtime is achieved more broadly whenever the weight is at least $\sqrt{M}$. Furthermore, the algorithm is also deterministic when the weight scales between $\sqrt{M}$ and $M$, and this is the first example of quantum search on the simplex of complete graphs that is both asymptotically optimal and deterministic. In addition, for weights where the algorithm is nondeterministic, we give a way to find the marked vertex by inspecting neighboring vertices. Finally, while it is known that connectivity is not a reliable indicator of fast quantum search when comparing different graph families, we show that it is also unreliable within the graph family of weighted simplex of complete graphs.
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.
The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$, and that the first counterexample occurs at $n=15$. The blow-up of the five cycle by independent sets of size three, equivalently the circulant on fifteen vertices with connection set $\{1,4,6\}$, has likelihood $0.20128\ldots$ times that of $K_{7,8}$, and it is again triangle-free. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(n\,2^n)$ and which closes on the blow-ups of any fixed base graph.
Let $\mathcal{B}$ be the class consisting of the six-vertex bipartite graphs that possess a perfect matching and their complements. It is proved that every $\mathcal{B}$-free graph $G$ satisfies $\alpha(G)+\omega(G)\ge |V(G)|-1$. This establishes Conjecture 3.1 of Litjens, Polak and Sivaraman (B-Free Graphs Conjecture). For a smallest counterexample, Hall-type exchange arguments show that two maximum stable sets, and likewise two maximum cliques, differ in at most two vertices. A core-corona matching lemma then forces $|\alpha(G)-\omega(G)|\le 2$. Double counting between suitably dense and sparse vertices reduces the problem to twenty-one binary feasibility systems on at most fourteen vertices. Their infeasibility is verified by two independent exact encodings, with a separate exhaustive validation of the forbidden-family constraints.
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, equality for $s^+$ holds exactly for trees, whereas equality for $s^-$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.
We develop a framework for proving universality results in sparse random graphs. As a first application, we show that there exists an absolute constant $C>1$ such that, with high probability, for every fixed constant $\Delta$, the binomial random graph $G(n,C\ln n/n)$ contains every $n$-vertex tree with maximum degree at most $\Delta$. This answers a question of Montgomery (Advances in Mathematics, 2019). We also determine, for every $p$ satisfying $C\ln n/n\leq p=n^{-1+o(1)}$, the minimum girth $\ell$ (up to an absolute multiplicative constant) for which with high probability $G(n,p)$ contains all cycle factors of girth at least $\Omega(\ell)$. In particular, with high probability $G(n,C\ln n/n)$ contains all cycle factors of girth at least $100\ln n/\ln\ln n$, which is optimal up to a constant factor. This extends an earlier result of Ferber, Kronenberg, and Luh (Transaction of the American Mathematical Society, 2019) and significantly improves a corollary of a deep result of Kahn, Lubetzky, and Wormald (Communications on Pure and Applied Mathematics, 2017). One of the key ingredients in the proofs is establishing the optimal depth of linking systems in sparse random graphs.
Asaf Cohen Antonir, Lyuben Lichev, Maksim Zhukovskii· 0 citations
For every odd integer $t\ge17$, we prove that an explicit circulant graph on $5t-10$ vertices is doubly saturated $R(3,t)$-good. The graph is triangle-free and has independence number $t-1$. Adding any nonedge creates a triangle, whereas deleting any edge creates an independent set of order $t$. This settles Conjecture 2 of Przybocki, Mackey, Heule, and Subercaseaux. A cyclic sumset identity and explicit witnesses prove the local saturation properties. Writing $t=2m+1$, a five-layer reduction proves the independence bound via a uniform affine certificate for $m\ge30$ and an exhaustive checker for $8\le m\le29$. The checker soundness and the complete argument are formalized in Lean 4.32.2. Consequently, $2t-1\le \operatorname{DS}(3,t)\le 5t-10$ for odd $t\ge17$.
Abhishek Saigal, Akaash R. Parthasarathy· 0 citations