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Let $\Gamma$ be a graph with diameter at least two. Then $\Gamma$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of $\Gamma$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume $\Gamma$ is $1$-homogeneous distance-regular with intersection number $a_1>0$ and $D\geqslant 5$. Define $b=b_1/(\theta_1+1)$, where $b_1$ is the intersection number and $\theta_1$ is the second largest eigenvalue of $\Gamma$. We show that if intersection number $c_2\geqslant 2$, then $b\geqslant 1$ and one of the following (i)--(vi) holds: (i) $\Gamma$ is a regular near $2D$-gon, (ii) $\Gamma$ is a Johnson graph $J(2D,D)$, (iii) $\Gamma$ is a halved $\ell$-cube where $\ell \in \{2D,2D+1\}$, (iv) $\Gamma$ is a folded Johnson graph $\bar{J}(4D,2D)$, (v) $\Gamma$ is a folded halved $(4D)$-cube, (vi) the valency of $\Gamma$ is bounded by a function of $b$. Using this result, we characterize $1$-homogeneous graphs with classical parameters and $a_1>0$, as well as tight distance-regular graphs., Comment: 19 pages

An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that an association scheme in which at most one relation is not strongly regular of (negative) Latin square type, is amorphic.

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 large with respect to $\lambda$., Comment: arXiv admin note: text overlap with arXiv:2011.11935

In this paper we study when the $q$-distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3., Comment: 15 pages

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 an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $\Gamma$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $\Gamma$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $\Gamma$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $\Gamma$ is bounded by a function in the parameter $b=b_1/(1+\theta_1)$, where $b_1$ is the intersection number of $\Gamma$ and $\theta_1$ is the second largest eigenvalue of $\Gamma$., Comment: 18 pages

In this paper, we study the $q$-distance matrix for a distance-regular graph and show that the $q$-distance matrix of a distance-regular graph with classical parameters ($D, q, \alpha, \beta$) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose $q$-distance matrix has exactly one positive eigenvalue., Comment: 19 pages

Graham and Lov\'{a}sz conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order $n$ is unimodal with the maximum value occurring at $\lfloor\frac{n}{2}\rfloor$. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.

Let $\Gamma$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$. Then $\Gamma$ is tight in the sense of Juri\v{s}i\'{c}, Koolen, and Terwilliger [12] whenever $\Gamma$ is locally strongly regular with nontrivial eigenvalues $p:=\theta_2$ and $-q:=\theta_3$. Assume that $\Gamma$ is tight. Then the intersection numbers of $\Gamma$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $\Gamma$. We denote $\Gamma$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$., Comment: 15 pages, 1 table

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs, and using Jacobi sums, we show that our construction produces infinitely many instances. Moreover, we prove some necessary conditions for the existence of (strictly) Neumaier graphs that allow us to show that several parameter sets are not admissible.

In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$., Comment: 13 pages, 2 tables

Koolen et al. showed that if a graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will focus on the sesqui-regular graphs with smallest eigenvalue at least $-3$ and study their integrability.

In this paper we give two characterizations of the $p \times q$-grid graphs as co-edge-regular graphs with four distinct eigenvalues., Comment: 18 pages, 1 table

In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal., Comment: 18 pages

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix to the $i$th subconstituent space of $\Gamma$ with respect to $x$. The Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), E^*_1(x), \ldots, E^*_D(x)$. Let $V$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $\Gamma$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, there exists a $\mathbb{C}$-vector space isomorphism $\rho:V\to V$ such that $(\rho A - A \rho)V=0$ and $(\rho E^*_i(x) - E^*_i(y)\rho)V=0$ for all $0\leq i \leq D$. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. For $D=2$, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For $D = 3$, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For $D=4$, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive., Comment: 30 pages

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 if a strongly regular graph contains a Delsarte clique, then the parameter $\mu$ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters $(v,k,\lambda,\mu)$ for strongly regular graphs. Lastly, we provide tables of parameters $(v,k,\lambda,\mu)$ for nonexistent strongly regular graphs with smallest eigenvalue $-4, -5, -6$ or $-7$.

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 related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest 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 eigenvalue and present some new findings.

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform hypergraph whose second eigenvalue is bounded by a given value. Our results improve and extend previous work done by Feng-Li (1996) on Alon-Boppana theorems for regular hypergraphs and by Dinitz-Schapira-Shahaf (2020) on the Moore or degree-diameter problem. We also determine the largest order of an $r$-regular $u$-uniform hypergraph with second eigenvalue at most $\theta$ for several parameters $(r,u,\theta)$. In particular, orthogonal arrays give the structure of the largest hypergraphs with second eigenvalue at most $1$ for every sufficiently large $r$. Moreover, we show that a generalized Moore geometry has the largest spectral gap among all hypergraphs of that order and degree., Comment: references updated, fixed some typos, added explanation describing the differences between the graphs and hypergraphs results, 27 pages, 3 tables, European Journal of Combinatorics, accepted for publication

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.

Let $\Gamma$ be a $Q$-polynomial distance-regular graph of diameter $d\geq 3$. Fix a vertex $\gamma$ of $\Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $\Gamma$ with respect to $\gamma$. We prove that this subgraph is connected., Comment: Journal of Combinatorial Theory, Series A, accepted for publication, 6 pages

In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples which do not arise in this manner, and we term these {\it pseudo-generalized quadrangles}. If the graph is a generalized quadrangle then $t \leq s^2$ and $s \leq t^2$, while for pseudo-generalized quadrangles we still have the former bound but not the latter. Previously, Neumaier has proved a bound for $s$ which is cubic in $t$, but we improve this to one which is quadratic. The proof involves a careful analysis of cliques and cocliques in the graph. This improved bound eliminates many potential parameter sets which were otherwise feasible.

A regular graph is co-edge regular if there exists a constant $\mu$ such that any two distinct and non-adjacent vertices have exactly $\mu$ common neighbors. In this paper, we show that for integers $s\ge 2$ and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$., Comment: arXiv admin note: text overlap with arXiv:1806.03593

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., Comment: 22 pages, 4 figures, submitted to The Art of Discrete and Applied Mathematics, https://adam-journal.eu/index.php/ADAM

In this note, we study the maximum number $N_\alpha(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $\alpha$, where $\frac{1}{\alpha}$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dr\"{a}xler, Keevash and Sudakov., Comment: 4 pages

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 study a new class of regular graphs called sesqui-regular graphs, which contains strongly regular graphs as a subclass, and prove that for a sesqui-regular graph with parameters $(v,k,c)$ and smallest eigenvalue at least $-\lambda$, if $k$ is very large, then either $c \leq \lambda^2(\lambda -1)$ or $v-k-1 \leq \frac{(\lambda-1)^2}{4} + 1$ holds.

In this paper we will show that there does not exist a distance-regular graph $\Gamma$ with intersection array $\{80, 54,12; 1, 6, 60\}$. We first show that a local graph $\Delta$ of $\Gamma$ does not contain a coclique with 5 vertices, and then we prove that the graph $\Gamma$ is geometric by showing that $\Delta$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $\Gamma$, and we could obtain that there is no such a distance-regular graph., Comment: 15 pages

Let $v(k, \lambda)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\lambda$. The Alon-Boppana Theorem implies that $v(k, \lambda)$ is finite when $k > \frac{\lambda^2 + 4}{4}$. In this paper, we show that for fixed $\lambda \geq1$, there exists a constant $C(\lambda)$ such that $2k+2 \leq v(k, \lambda) \leq 2k + C(\lambda)$ when $k > \frac{\lambda^2 + 4}{4}$., Comment: 13 pages

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 $n-(2-o(1))\log_2 n$., Comment: 18 page, 24 tables, accepted for publication to Experimental Mathematics

Koolen et al. showed that if a connected graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.

In this paper we show that for integers $s\geq2$, $t\geq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $t\times t$-grid is the $s$-clique extension of the $t\times t$-grid, if $t$ is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough., Comment: 13 pages

In 1995, Metsch showed that the Grassmann graph $J_q(n,D)$ of diameter $D\geq 3$ is characterized by its intersection numbers with the following possible exceptions: (-) $n=2D$ or $n=2D+1$, $q\geq 2$; (-) $n=2D+2$ and $q\in \{2,3\}$; (-) $n=2D+3$ and $q=2$. In 2005, Van Dam and Koolen constructed the twisted Grassmann graphs with the same intersection numbers as the Grassmann graphs $J_q(2D+1,D)$, for any prime power $q$ and diameter $D\geq 2$, but they are not isomorphic. We show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection numbers provided that the diameter $D$ is large enough.

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, 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 $g\geq 2d-2$., Comment: 24 pages, 7 tables; In the revised version, a new result (Theorem 4.12) is added at the end of Section 4 showing that the bipartite bound from this paper (Theorem 4.7) is always better than the general upper bound from our previous SIDMA paper (Theorem 4.9)

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.

We exhibit infinitely many examples of edge-regular graphs that have regular cliques and that are not strongly regular. This answers a question of Neumaier from 1981., Comment: 11 pages; accepted to the European Journal of Combinatorics

An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete $4$-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known $2$-arc-transitive covers of the cube: the M\"{o}bius-Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive $2$-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three., Comment: 26 pages, 12 figures

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, introduced by Woo and Neumaier in 1995. Our result says that for every $\lambda \leq -2$, if a graph with smallest eigenvalue at least $\lambda$ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph $H(3,q)$, the Johnson graph $J(v, 3)$ and the $2$-clique extension of grids, respectively., Comment: 26 pages

In this paper, we determine the integrally representable trees of norm 3., Comment: 18 pages

In this paper, we introduce the concepts of the plain eigenvalue, the main-plain index and the refined spectrum of graphs. We focus on the graphs with two main and two plain eigenvalues and give some characterizations of them., Comment: 9 pages