Related papers: Sesqui-regular graphs with fixed smallest eigenval…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs…

In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2/k<=C. This extends a conjecture of Bannai and Ito.

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of all complex irreducible characters of $G$. The character-graph $\Delta(G)$ associated to $G$, is a graph whose vertex set is the set of primes which divide the degrees of some…

Let $G$ be a connected undirected graph with $n$, $n\ge 3$, vertices and $m$ edges. Denote by $\rho_1 \ge \rho_2 \ge \cdots > \rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $\rho_i$, $i=1,2,\ldots , n-1$,…

In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penji\' c in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier…

A Deza graph $G$ with parameters $(n,k,b,a)$ is a $k$-regular graph with $n$ vertices such that any two distinct vertices have $b$ or $a$ common neighbours. The children $G_A$ and $G_B$ of a Deza graph $G$ are defined on the vertex set of…

We prove that for sufficiently large k, there exist $0\le\sigma_k\le\eps_k\to 0$ as $k\to\infty$, such that asymptotically almost surely the first k-regular subgraph appeared in the random graph process where one edge is added at a time has…

The $n$-th Fiedler value of a class of graphs $\mathcal C$ is the maximum second eigenvalue $\lambda_2(G)$ of a graph $G\in\mathcal C$ with $n$ vertices. In this note we relate this value to shallow minors and, as a corollary, we determine…

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order.…

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…

In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well…

Let $\Sigma$ be a surface with boundary $b(\Sigma)$, $\mathcal{L}$ be a collection of $k$ disjoint $b(\Sigma)$-paths in $\Sigma$, and $P$ be a non-separating $b(\Sigma)$-path in $\Sigma$. We prove that there is a homeomorphism $\phi: \Sigma…

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph…

Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and…

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the…

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard…

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends…

We show that the size-Ramsey number of any cubic graph with $n$ vertices is $O(n^{8/5})$, improving a bound of $n^{5/3 + o(1)}$ due to Kohayakawa, R\"{o}dl, Schacht, and Szemer\'{e}di. The heart of the argument is to show that there is a…