Recent Progress on Graphs with Fixed Smallest Adjacency Eigenvalue: A Survey

Abstract

We give a survey on graphs with fixed smallest adjacency 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 related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest adjacency eigenvalue and large minimal valency;

• (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest adjacency eigenvalue and the characterizations of certain families of distance-regular graphs.

At the end of the survey, we also discuss signed graphs with fixed smallest adjacency eigenvalue and present some new findings.

Appendix A. Q-polynomial distance-regular graphs

Let V denote a non-empty finite set. Let $\mathrm {Mat}_V({\mathbb {C}})$ denote the ${\mathbb {C}}$-algebra consisting of all complex matrices whose rows and columns are indexed by V. Let $\mathbb {U}={\mathbb {C}}^V$ denote the ${\mathbb {C}}$-vector space consisting of all complex vectors indexed by V. We endow $\mathbb {U}$ with standard Hermitian inner product $({\mathbf {u}},{\mathbf {v}})={\mathbf {u}}^T\overline{{\mathbf {v}}}$ for ${\mathbf {u}},{\mathbf {v}} \in \mathbb {U}$. We view $\mathbb {U}$ as a left module for $\mathrm {Mat}_V({\mathbb {C}})$, called the standard module.

Let G be a distance-regular graph of diameter D. Let V be the vertex set of G. For $0\le i\le D$, let $A_i$ denote the matrix in $\mathrm {Max}_{V}({\mathbb {C}})$ defined by

$$\begin{aligned} (A_i)_{x,y}=\left\{ \begin{array}{ll} 1 &{} \text { if }d(x,y)=i, \\ 0 &{} \text { otherwise}, \end{array} \right. \end{aligned}$$

where $x,y\in V$. We call $A_i$ the ith distance matrix of G. We abbreviate $A:=A_1$. Observe that

• (1a) $A_0=I$;

• (1b) $\sum ^D_{i=0}A_i=J$, the all-ones matrix;

• (1c) each $A_i$ is real symmetric;

• (1d) there exist $p_{ij}^h$ for $0\le i,j,h\le D$, such that $A_iA_j =A_jA_i= \sum ^D_{h=0}p^h_{ij}A_h$ hold.

Notice that (1a) implies for each pair vertices $x,y\in V$ with $d(x,y)=h$, the equality $|G_i(x)\cap G_j(y)|=p_{ij}^h$ holds. Therefore, for all integers $0\le h,i,j\le D$, $p^h_{ij}=0$ (resp. $p^h_{ij}\ne 0$) if one of h, i, j is greater than (resp. equal to) the sum of the other two. By these facts, we find that $A_0, A_1,\ldots , A_D$ is a basis for a commutative subalgebra M of $\mathrm {Mat}_V({\mathbb {C}})$, which we call the Bose-Mesner algebra of G. It is known that A generates M, as $AA_i=c_{i+1}A_{i+1}+a_iA_i+b_{i-1}A_{i-1}$ ($0\le i\le D$) by condition (iv), where $\{b_0,b_1,\ldots ,b_{D-1};c_1,c_2,\ldots ,c_D\}$ is the intersection array of G.

The algebra M has a second basis $E_0, E_1, \ldots , E_D$ such that

• (2a) $E_0=|V|^{-1}J$,

• (2b) $\sum ^D_{i=0}E_i=I$,

• (2c) each $E_i$ is real symmetric,

• (2d) $E_iE_j=E_jE_i=\delta _{ij}E_i$

(see [12, p. 45]). We call $E_i$ the ith primitive idempotent of G. Since $\{E_i\}^D_{i=0}$ is a basis for M, there exist complex scalars $\{\theta _i\}^D_{i=0}$ such that $A=\sum ^D_{i=0}\theta _i E_i$. (Note that $\{\theta _i\}^D_{i=0}$ are exactly all of the distinct eigenvalues of G and they are real.) Observe $AE_i=E_iA=\theta _iE_i$ for $0\le i \le D$. We call $\theta _i$ the eigenvalue of G associated with $E_i$ for $0\le i \le D$. Observe $\mathbb {U}=E_0\mathbb {U}\oplus E_1\mathbb {U}\oplus \cdots \oplus E_D\mathbb {U}$, an orthogonal direct sum. For $0\le i \le D$, $E_i\mathbb {U}$ is the eigenspace of A associated with $\theta _i$. Denote by $m_i$ the rank of $E_i$ and observe $m_i=\dim (E_i\mathbb {U})$, the multiplicity of the eigenvalue $\theta _i$.

We now introduce the notion of Q-polynomial property of G. Let $\circ$ denote the entrywise product in $\mathrm {Mat}_V({\mathbb {C}})$. Since $A_i\circ A_j =\delta _{ij}A_i$, the Bose-Mesner algebra M is closed under $\circ$. Also as $\{E_i\}^D_{i=0}$ is a basis for M, there exist complex scalars $q^h_{ij}$ such that

$$\begin{aligned} E_i\circ E_j = |V|^{-1}\sum ^D_{h=0}q^h_{ij}E_h. \end{aligned}$$

By [12, p. 48, p. 49], the scalars $q^h_{ij}$ are real and non-negative. We say G is Q-polynomial (with respect to the given ordering $E_0, E_1, \ldots , E_D$) whenever for all integers $0\le h,i,j\le D$, $q^h_{ij}=0$ (resp. $q^h_{ij}\ne 0$) if one of h, i, j is greater than (resp. equal to) the sum of the other two [12, p. 235].

We assume G is Q-polynomial with respect to the ordering $E_0, E_1, \ldots , E_D$. Fix a vertex $x \in V$. We refer to x as a “base” vertex. For $0 \le i \le D$, we define the diagonal matrix $E^*_i=E^*_i(x) \in \mathrm {Mat}_V({\mathbb {C}})$ with diagonal entry

$$\begin{aligned} (E^*_i)_{y,y} =\left\{ \begin{array}{ll} 1 &{} \text { if }d(x,y)=i, \\ 0 &{} \text { otherwise}, \end{array} \right. \end{aligned}$$

where $y\in V$. We call $E^*_i$ the ith dual primitive idempotent of G with respect to x. Observe

• (3a) $\sum ^D_{i=0}E^*_i=I$,

• (3b) each $E^*_i$ is real symmetric,

• (3c) $E^*_iE^*_j=\delta _{ij}E^*_i$.

By these facts, $E^*_0, E^*_1, \ldots , E^*_D$ is a basis for a commutative subalgebra $M^*$ of $\mathrm {Mat}_V({\mathbb {C}})$, which we call the dual Bose-Mesner algebra of G.

Define the diagonal matrix $A^*_i=A^*_i(x) \in \mathrm {Mat}_V({\mathbb {C}})$ with diagonal entry $(A^*_i)_{y,y} = |V|(E_i)_{x,y}$ for $y\in V$. By [89, p. 379], $A^*_0, A^*_1, \ldots , A^*_D$ is also a basis for $M^*$, and moreover

• (4a) $A^*_0=I$,

• (4b) $\sum ^D_{i=0}A^*_i=|V|E_0^*$,

• (4c) each $A_i^*$ is real and symmetric,

• (4d) $A^*_iA^*_j=A^*_jA^*_i=\sum ^D_{h=0} q^h_{ij} A^*_h$.

We call $A^*_i$ the ith dual distance matrix of G with respect to x. We abbreviate $A^*=A^*_1$, called the dual adjacency matrix of G with respect to x. From conditions (4a) and (4d), we find that the matrix $A^*$ generates $M^*$. Since $\{E^*_i\}^D_{i=0}$ is a basis for $M^*$, there exist complex scalars $\{\theta ^*_i\}^D_{i=0}$ such that $A^*=\sum ^D_{i=0}\theta ^*_iE^*_i$. Observe $A^*E^*_i=E^*_iA^*=\theta ^*_iE^*_i$ for $0 \le i \le D$. The scalars $\{\theta ^*_i\}^D_{i=0}$ are real [89, Lemma 3.11] and mutually distinct. We call $\theta ^*_i$ the dual eigenvalue of G associated with $E^*_i$. Observe $\mathbb {U}=E^*_0\mathbb {U}\oplus E^*_1\mathbb {U}\oplus \cdots \oplus E^*_D\mathbb {U}$, an orthogonal direct sum. For $0 \le i \le D$, the space $E^*_i\mathbb {U}$ is the eigenspace of $A^*$ associated with $\theta ^*_i$.

Let $T=T(x)$ denote the subalgebra of $\mathrm {Mat}_V({\mathbb {C}})$ generated by M and $M^*$. We call T the Terwilliger algebra (or subconstituent algebra) of G with respect to x [89]. Note that A and $A^*(x)$ generates T. The algebra T is finite dimensional and non-commutative. It is also semi-simple since it is closed under conjugate and transpose map. The following are relations in T [89, Lemma 3.2]. For $0 \le h,i,j \le D$,

$$\begin{aligned} E^*_iA_hE^*_j =&0 \quad \text {if and only if} \quad p^h_{ij}=0,\\ E_iA^*_hE_j =&0 \quad \text {if and only if} \quad q^h_{ij}=0. \end{aligned}$$

Note that T may depend on the choice of the base vertex (see [8]).

By a T-module, we mean a subspace $\mathbb {W}$ of $\mathbb {U}$ such that $B\mathbb {W}\subseteq \mathbb {W}$ for all $B\in T$. Observe that $\mathbb {U}$ is a T-module, called the standard module of T (or standard T-module). A T-module is called irreducible if it contains no T-submodule except itself and zero module.

Let $\mathbb {W}$ be a T-module and $\mathbb {W}_1$ a T-submodule of $\mathbb {W}$. Then the orthogonal complement of $W_1$ in W is a T-module, since T is closed under conjugate transpose map. It follows that W decomposes into an orthogonal direct sum of irreducible T-modules.

Let W denote an irreducible T-module. Then W decomposes into a direct sum of nonzero spaces among $E^*_iW$, $0 \le i \le D$. By the endpoint of W, we mean $\min \{i \mid 0 \le i \le D, E^*_iW\ne 0\}$. By the diameter of W, we mean $|\{i \mid 0 \le i \le D, E^*_iW\ne 0\}|-1$. Let r denote the endpoint of W and d the diameter of W. By [89, Lemma 3.9], we have (i) $E^*_iW \ne 0$ if and only if $r \le i \le r+d$; (ii) $W=\bigoplus ^d_{h=0}E^*_{r+h}W$, an orthogonal direct sum. An irreducible T-module W is said to be thin whenever $\dim (E^*_iW)\le 1$ for $0 \le i \le D$. There exists a unique thin irreducible T-module with endpoint 0 and diameter D, which we call it the primary T-module. The primary T-module has a basis $E^*_0{\mathbf {j}}, \ldots , E^*_D{\mathbf {j}}$ [89, Lemma 3.6], where ${\mathbf {j}}$ is the all-ones vector.

The graph G is said to be thin with respect to x whenever every irreducible T(x)-module is thin. The graph G is said to be thin whenever G is thin with respect to every vertex x of G. See [91, Section 6] for examples of thin Q-polynomial distance-regular graphs.