Combinatorial threshold-linear networks (CTLNs) are a special class of neural networks whose dynamics are tightly controlled by an underlying directed graph. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics [1], and we conjectured that these are the only stable fixed points allowed [2]. In this paper we prove that the conjecture holds in a variety of special cases, including for graphs with very strong inhibition and graphs of size n ≤ 4. We also provide further evidence arXiv:1909.02947v1 [q-bio.NC] 27 Aug 2019 def = {1, . . . , n} which have positive firing rate (see Section 2.1 for more details). In prior work [1], we proved a series a graph rules showing that we can rule in and rule out fixed points of W solely from features of the corresponding graph. We also conjectured that the supports of stable fixed points of CTLNs precisely correspond to target-free cliques. These are subsets σ ⊆ [n] that are cliques (all-to-all bidirectionally connected in G) that do not have targets. A node k is a target of σ if k / ∈ σ and i → k for all i ∈ σ. Conjecture 1.3 ([2]). Let W = W (G, ε, δ) be a CTLN on n nodes with graph G, and let σ ⊆ [n]. Then σ is the support of a stable fixed point if and only if σ is a target-free clique.

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero-one matrix A avoids another d-dimensional zero-one matrix P if no sub-matrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the maximum number of nonzero entries in a d-dimensional n × · · · × n matrix that avoids P. This maximum number, denoted by f (n, P, d), is called the extremal function. We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on f (n, P, d) when n is large for every d-dimensional block permutation matrix P. We establish the tight bound Θ(n d−1) on f (n, P, d) for every d-dimensional tuple permutation matrix P. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that f (n, P, d) is super-homogeneous for a class of matrices P. We use this super-homogeneity to show that the limit inferior of the sequence { f (n,P,d) n d−1 } has a lower bound 2 Ω(k 1/d) for a family of k × · · ·× k permutation matrices P. We also improve the upper bound on the limit superior from 2 O(k log k) to 2 O(k) for all k × · · · × k permutation matrices and show that the new upper bound also holds for tuple permutation matrices.

An (r, s)-formation is a concatenation of s permutations of r letters. If u is a sequence with r distinct letters, then let Ex (u, n) be the maximum length of any r-sparse sequence with n distinct letters which has no subsequence isomorphic to u. For every sequence u define fw (u), the formation width of u, to be the minimum s for which there exists r such that there is a subsequence isomorphic to u in every (r, s)-formation. We use fw (u) to prove upper bounds on Ex (u, n) for sequences u such that u contains an alternation with the same formation width as u.We generalize Nivasch's bounds on Ex ((ab) t , n) by showing that fw ((12 . . . l) t ) = 2t−1 and Ex ((12 . . . l) t , n) = n2by the NSF Graduate Research Fellowship under Grant No. 1122374.obtained from v by only moving the first letter of v to another place in v, then we show that fw (u) = 4 and Ex (u, n) = Θ(nα(n)). Furthermore we prove that fw (abc(acb) t ) = 2t + 1 and Ex (abc(acb) t , n) = n2 1 (t−1)! α(n) t−1 ±O(α(n) t−2 ) for every t ≥ 2.

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero-one matrix A avoids another d-dimensional zero-one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to study the maximum number of nonzero entries in a d-dimensional n × · · · × n matrix that avoids P. This maximum number, denoted by f (n, P, d), is called the extremal function.We advance the extremal theory of matrices in two directions. The methods that we use come from combinatorics, probability, and analysis. Firstly, we obtain non-trivial lower and upper bounds on f (n, P, d) when n is large for every d-dimensional block permutation matrix P. We establish the tight bound Θ(n d−1 ) on f (n, P, d) for every d-dimensional tuple permutation matrix P. This tight bound has the lowest possible order that an extremal function of a nontrivial matrix can ever achieve. Secondly, we show that f (n, P, d) is super-homogeneous for a class of matrices P. We use this super-homogeneity to show that the limit inferior of the sequence { f (n,P,d) n d−1 } has a lower bound 2 Ω(k 1/d ) for a family of k × · · ·× k permutation matrices P. We also improve the upper bound on the limit superior from 2 O(k log k) to 2 O(k) for all k × · · · × k permutation matrices and show that the new upper bound also holds for tuple permutation matrices.

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We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semi-bar k-visibility graphs between 2 3 (k + 1) and 2k. In addition we show that the maximum number of edges in arc and circle k-visibility graphs on n vertices is at most (k + 1)(3n − k − 2) for n > 4k +4 and n 2 for n ≤ 4k +4, while the maximum chromatic number is at most 6k + 6. In semi-arc k-visibility graphs on n vertices, we show that the maximum number of edges is n 2 for n ≤ 3k + 3 and at most (k + 1)(2n − k+2 2 ) for n > 3k + 3, while the maximum chromatic number is at most 4k + 4. arXiv:1305.0505v2 [math.CO]

In this paper we initiate the study of broadcast dimension, a variant of metric dimension. Let G be a graph with vertex set V (G), and let d(u, w) denote the length of a u − w geodesic in G.over all resolving broadcasts of G, where c f (G) can be viewed as the total cost of the transmitters (of various strength) used in resolving the entire network described by the graph G. Note that bdim(G) reduces to adim(G) (the adjacency dimension of G, introduced by Jannesari and Omoomi in 2012) if the codomain of resolving broadcasts is restricted to {0, 1}. We determine its value for cycles, paths, and other families of graphs. We prove that bdim(G) = Ω(log n) for all graphs G of order n, and that the result is sharp up to a constant factor. We show that adim (G) bdim(G) and bdim(G) dim(G) can both be arbitrarily large, where dim(G) denotes the metric dimension of G. We also examine the effect of vertex deletion on the adjacency dimension and the broadcast dimension of graphs.

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