Related papers: A Bose-Laskar-Hoffman theory for $\mu$-bounded gra…

We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory…

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least $-2$. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph,…

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda$ can be defined by a finite set of forbidden induced subgraphs if and only…

One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with $\mu\geq 2$ and second largest eigenvalue $\theta_1= b_1-1$. In this paper we give a classification under the…

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann…

In 1979, Neumaier gave a bound on $\lambda$ in terms of $m$ and $\mu$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular…

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest…

In this paper, we show that a connected graph with smallest eigenvalue at least -3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least -2.

A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very…

In this paper, we study the order of a maximal clique in an amply regular graph with a fixed smallest eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that…

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this…

We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we…

In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular…

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the…

Let $\lambda\geq2$ be an integer. For strongly regular graphs with parameters $(v, k, a,c)$ and smallest eigenvalue $-\lambda$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. In this paper, we will…

In this paper we will give a structure theory for regular graphs with fixed smallest eigenvalue. As a consequence of this theory, we show that a $k$-regular graph with smallest eigenvalue $-\lambda$ has clique number linear in $k$ if $k$ is…

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower…

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$. Juri\v{s}i\'c and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the block graph of…

We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the…

We introduce a new family of closed differential forms naturally associated with minimal graphical submanifolds in Euclidean space, defined in arbitrary codimension. For each minimal graph, we construct an explicit closed form whose…