An official website of the United States government
The spread of a graph $$G$$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $$G$$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $$n$$, the $$n$$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $$\lceil(2n-1)/3\rceil$$ vertices and $$\lfloor(n-2)/3\rfloor$$ isolated vertices. For planar graphs, we show that the extremal $$n$$-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on $$\lceil(2n-2)/3\rceil$$ vertices and $$\lfloor(n-4)/3\rfloor$$ isolated vertices.
more »
« less
This content will become publicly available on January 17, 2026
Maximum Spread of $K_{s,t}$-Minor-Free Graphs
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 $$t \geq \frac32(s-3) +\frac{4}{s-1}$$.
more »
« less
- Award ID(s):
- 2038080
- PAR ID:
- 10615621
- Publisher / Repository:
- https://www.combinatorics.org/
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 32
- Issue:
- 1
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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.more » « less
-
Let $$G$$ be a $$t$$-tough graph on $$n\ge 3$$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $$G$$ is greater than $$\frac{n}{t+1}-1$$, then $$G$$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $$t$$ is between 1 and 2, and recently the second author proved a general result. The result states that if the degree sum of any two nonadjacent vertices of $$G$$ is greater than $$\frac{2n}{t+1}+t-2$$, then $$G$$ is hamiltonian. It was conjectured in the same paper that the $+t$ in the bound $$\frac{2n}{t+1}+t-2$$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $$t$$-tough graphs $$G$$ on $$n\ge 3$$ vertices for which $$\sigma_2(G) = \frac{2n}{t+1}-2$$ but $$G$$ is non-hamiltonian.more » « less
-
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 .more » « less
-
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.more » « less
