arXiv subjects
Publications and source records attributed to Sebastian M. Cioabă.
In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana theorems for regular graphs and in the classical degree-diameter/Moore problem.
In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.
Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.
Let $b(k,θ)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $θ$. In this paper, we obtain a general upper bound for $b(k,θ)$ for any $0\leq θ< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,θ)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $θ$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graphs with $g\geq 2d-2$.
Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))\log_2 n$.
Let $Γ$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(Γ,T)$ be a Cayley graph of $Γ$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{τ\in T}|\mathrm{supp}(τ)|\leq 5$, where $\mathrm{supp}(τ)$ is the set of points in $[n]$ non-fixed by $τ$.
A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a path of length $\ell-1$ which is a subgraph of a complete graph $K_n$. Cámara and Haemers~\cite{MC} conjectured that $K_n \backslash P_{\ell}$ is determined by its adjacency spectrum for every $2\leq \ell \leq n$. In this paper we show that the conjecture is true for $7\leq \ell \leq9$.
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 eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes.
Let $k\ge 3$ be an integer, $q$ be a prime power, and $\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\in\mathbb{F}_q[X]$, $3\le i\le k$, such that $g_i(-X) = -\, g_i(X)$. We define a graph $S(k,q) = S(k,q;f_3,g_3,\cdots,f_k,g_k)$ as a graph with the vertex set $\mathbb{F}_q^k$ and edges defined as follows: vertices $a = (a_1,a_2,\ldots,a_k)$ and $b = (b_1,b_2,\ldots,b_k)$ are adjacent if $a_1\ne b_1$ and the following $k-2$ relations on their components hold: $$ b_i-a_i = g_i(b_1-a_1)f_i\Bigl(\frac{b_2-a_2}{b_1-a_1}\Bigr)\;,\quad 3\le i\le k. $$ We show that graphs $S(k,q)$ generalize several recently studied examples of regular expanders and can provide many new such examples.
Let $n\geq k\geq 2$ be two integers and $S$ a subset of $\{0,1,\dots,k-1\}$. The graph $J_{S}(n,k)$ has as vertices the $k$-subsets of the $n$-set $[n]=\{1,\dots,n\}$ and two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|\in S$. In this paper, we use Godsil-McKay switching to prove that for $m\geq 0$, $k\geq \max(m+2,3)$ and $S = \{0, 1, ..., m\}$, the graphs $J_S(3k-2m-1,k)$ are not determined by spectrum and for $m\geq 2$, $n\geq 4m+2$ and $S = \{0,1,...,m\}$ the graphs $J_{S}(n,2m+1)$ are not determined by spectrum. We also report some computational searches for Godsil-McKay switching sets in the union of classes in the Johnson scheme for $k\leq 5$.
From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $λ<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $λ$. In this paper, we investigate the largest number of vertices of such graphs.
Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of matchings in distance-regular graphs. A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. We generalize previous results on strongly regular graphs and show that all distance-regular graphs with diameter $D\geq 3$ are $2$-extendable. We also obtain various lower bounds for the extendability of distance-regular graphs of valency $k$ that depend on $k$, $λ$ and $μ$, where $λ$ is the number of common neighbors of any two adjacent vertices and $μ$ is the number of common neighbors of any two vertices in distance two.
Graham and Pollak showed that the vertices of any connected graph $G$ can be assigned $t$-tuples with entries in $\{0, a, b\}$, called addresses, such that the distance in $G$ between any two vertices equals the number of positions in their addresses where one of the addresses equals $a$ and the other equals $b$. In this paper, we are interested in determining the minimum value of such $t$ for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
Many regular graphs admit a natural partition of their edge set into cliques of the same order such that each vertex is contained in the same number of cliques. In this paper, we study the mixing rate of certain random walks on such graphs and we generalize previous results of Alon, Benjamini, Lubetzky and Sodin regarding the mixing rates of non-backtracking random walks on regular graphs.
The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties.
Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we give two infinite families of graphs for which this is not true. We also compute $M(G)$ for various graphs, and outline some connections between our problem and the existence of designs of various types.
Let $q=p^e$, where $p$ is a prime and $e\geq 1$ is an integer. For $m\geq 1$, let $P$ and $L$ be two copies of the $(m+1)$-dimensional vector spaces over the finite field $\mathbb{F}_q$. Consider the bipartite graph $W_m(q)$ with partite sets $P$ and $L$ defined as follows: a point $(p)=(p_1,p_2,\ldots,p_{m+1})\in P$ is adjacent to a line $[l]=[l_1,l_2,\ldots,l_{m+1}]\in L$ if and only if the following $m$ equalities hold: $l_{i+1} + p_{i+1}=l_{i}p_1$ for $i=1,\ldots, m$. We call the graphs $W_m(q)$ Wenger graphs. In this paper, we determine all distinct eigenvalues of the adjacency matrix of $W_m(q)$ and their multiplicities. We also survey results on Wenger graphs.