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1. Maximum Spread of $$K_{s,t}$$-Minor-Free Graphs

   https://doi.org/10.37236/13410  

   Linz, William; Lu, Linyuan; Wang, Zhiyu (January 2025, The Electronic Journal of Combinatorics)

   The spread of a graph $$G$$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $$G$$. In this paper, we consider the family of graphs which contain no $$K_{s,t}$$-minor. We show that for any $$t\geq s \geq  2$$ and sufficiently large $$n$$, there is an integer $$\xi_{t}$$ such that the extremal $$n$$-vertex $$K_{s,t}$$-minor-free graph attaining the maximum spread is the graph obtained by joining a graph $$L$$ on $(s-1)$ vertices to the disjoint union of $$\lfloor \frac{2n+\xi_{t}}{3t}\rfloor$$ copies of $$K_t$$ and $$n-s+1 - t\lfloor \frac{2n+\xi_t}{3t}\rfloor$$ isolated vertices. Furthermore, we give an explicit formula for $$\xi_{t}$$ and an explicit description for the graph $$L$$ for $$t \geq \frac32(s-3) +\frac{4}{s-1}$$. 

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2. Maximum spread of graphs and bipartite graphs

   https://doi.org/10.1090/cams/14  

   Breen, Jane; Riasanovsky, Alex W.; Tait, Michael; Urschel, John (December 2022, Communications of the American Mathematical Society)

   Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms. 

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3. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

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4. On 2-fold graceful graphs

   Bunge, R; Cornett, M; El-Zanati, S; Jeffries, J; Rattin, E. (January 2020, Journal of Combinatorial Mathematics and Combinatorial Computing)

   null (Ed.)

   A long-standing conjecture by Kotzig, Ringel, and Rosa states that every tree admits a graceful labeling. That is, for any tree $$T$$ with $$n$$~edges, it is conjectured that there exists a labeling $$f\colon V(T) \to \{0,1,\ldots,n\}$$ such that the set of induced edge labels $$\bigl\{ |f(u)-f(v)| : \{u,v\}\in E(T) \bigr\}$$ is exactly $$\{1,2,\ldots,n\}$$. We extend this concept to allow for multigraphs with edge multiplicity at most~$$2$$. A \emph{2-fold graceful labeling} of a graph (or multigraph) $$G$$ with $$n$$~edges is a one-to-one function $$f\colon V(G) \to \{0,1,\ldots,n\}$$ such that the multiset of induced edge labels is comprised of two copies of each element in $$\bigl\{ 1,2,\ldots, \lfloor n/2 \rfloor \bigr\}$$ and, if $$n$$ is odd, one copy of $$\bigl\{ \lceil n/2 \rceil \bigr\}$$. When $$n$$ is even, this concept is similar to that of 2-equitable labelings which were introduced by Bloom and have been studied for several classes of graphs. We show that caterpillars, cycles of length $$n \not\equiv 1 \pmod{4}$$, and complete bipartite graphs admit 2-fold graceful labelings. We also show that under certain conditions, the join of a tree and an empty graph (i.e., a graph with vertices but no edges) is $$2$$-fold graceful. 

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5. Many Cliques with Few Edges

   https://doi.org/10.37236/9550  

   Kirsch, Rachel; Radcliffe, A. J. (January 2021, The Electronic Journal of Combinatorics)

   null (Ed.)

   Recently Cutler and Radcliffe proved that the graph on $$n$$ vertices with maximum degree at most $$r$$ having the most cliques is a disjoint union of $$\lfloor n/(r+1)\rfloor$$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $$m$$ edges and maximum degree at most $$r$$, the graph that has the most cliques of size at least two is the disjoint union of $$\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$$ cliques of size $r+1$ together with the colex graph using the remainder of the edges. 

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