Safet Penjić

You can download thesis from http://osebje.famnit.upr.si/~penjic/research/acdrg.pdf

Abstract:
Through this thesis we introduce distance-regular graphs, and present some of their characterizations which depend on information retrieved from their adjacency matrix, principal idempotent matrices, predistance polynomials and spectrum. Let \Gamma be a finite simple connected graph. In Chapter I we present some basic results from Algebraic graph theory: we prove Perron-Frobenius theorem, we show how to compute the number of walks of a given length between two vertices of \Gamma, how to compute the total number of (rooted) closed walks of a given length of \Gamma, we introduce adjaceny matrix \A of \Gamma, principal idempotent matrices \E_i of \Gamma and introduce adjacent (Bose-Mesner) algebra of \Gamma and Hoffman polynomial of \Gamma. All of these results are needed in Chapters II and III. In Chapter II we define distance-regular graphs, show some examples of these graph, introduce distance-i matrix \A_i, i=0,1,...,D (where D is the diameter of graph \Gamma), introduce predistance polynomials p_i, i=0,1,...,d (d is the number of distinct eigenvalues) of \Gamma and prove the following sequence of equivalences: \Gamma is distance-regular  \Longleftrightarrow  \Gamma is distance-regular around each of its vertices and with the same intersection array  \Longleftrightarrow  distance matrices of \Gamma satisfy \A_i\A_j=\sum_{k=0}^Dp_{ij}^k\A_k,\,(0 \le i,j \le D) for some constants p_{ij}^k  \Longleftrightarrow  for some constants a_h, b_h, c_h (0\le h \le D), c_0 = b_D = 0, distance matrices of \Gamma satisfy the three-term recurrence \A_h\A = b_{h-1}\A_{h-1} + a_h\A_h + c_{h+1}\A_{h+1}, (0 \le h \le D)  \Longleftrightarrow  \{I,\A,...,\A_D\} is a basis of the adjacency algebra \mathcal A(\Gamma)  \Longleftrightarrow  \A acts by right (or left) multiplication as a linear operator on the vector space \span\{I,\A_1,\A_2,...,\A_D\}  \Longleftrightarrow  for any integer h, 0\le h\le D, the distance-h matrix \A_h is a polynomial of degree h in \A  \Longleftrightarrow  \Gamma is regular, has spectrally maximum diameter (D=d) and the matrix \A_D is polynomial in \A  \Longleftrightarrow  the number a^\ell_{uv} of walks of length \ell between two vertices u, v \in V only depends on h = \partial(u,v)  \Longleftrightarrow  for any two vertices u, v\in V at distance h, we have a_{uv}^h = a_h^h and a_{uv}^{h+1} = a_h^{h+1} for any 0\le h\le D - 1, and a_{uv}^D = a_D^D for h = D  \Longleftrightarrow  \A_i\E_j=p_{ji}\E_j (p_{ji} are some constants) \Leftrightarrow \A_i=\sum_{j=0}^d p_{ji} \E_j \Leftrightarrow \A_i=\sum_{j=0}^d p_{i}(\lambda_j) \E_j \Leftrightarrow \A_i\in{\cal A}, (i,j=0,1,...,d(=D))  \Longleftrightarrow  for every 0 \le i \le d and for every pair of vertices u,v of \Gamma, the (u,v)-entry of \E_i depends only on the distance between u and v  \Longleftrightarrow  \E_j\circ\A_i=q_{ij}\A_i (q_{ij} are some constants) \Leftrightarrow \E_j=\sum_{i=0}^D q_{ij} \A_i \Leftrightarrow \E_j=\frac{1}{n}\sum_{i=0}^d q_{i}(\lambda_j) \A_i (where q_i(\lambda_j):=m_j\frac{p_i(\lambda_j)}{p_i(\lambda_0)}) \Leftrightarrow \E_j\in\mathcal {\cal D} i,j=0,1,...,d(=D)  \Longleftrightarrow  \A^j\circ \A_i=a_i^{(j)} \A_i (a_i^{(j)} are some constants) \Leftrightarrow \A^j=\sum_{i=0}^d a_i^{(j)} \A_i \Leftrightarrow \A^j=\sum_{i=0}^d\sum_{l=0}^d q_{i\ell}\lambda_l^j \A_i \Leftrightarrow \A^j\in\mathcal D i,j=0,1,...,d. Finally, in Chapter III, we introduce one interesting family of orthogonal polynomials - the canonical orthogonal system, and prove three more characterizations of distance-regularity which involve the spectrum: \Gamma is distance-regular  \Longleftrightarrow  the number of vertices at distance k from every vertex u \in V is |\Gamma_k(u)|=p_k(\lambda_0) for 0\le k\le d (where \{p_k\}_{0\le k\le d} are predistance polynomials)  \Longleftrightarrow  \displaystyle{ q_k(\lambda_0) = \frac{n}{ \sum_{u\in V} \frac{1}{s_k(u)} } } for 0\le k\le d (where q_k=p_0+...+p_k, s_k(u)=|\Gamma_0(u)|+|\Gamma_1(u)|+...+|\Gamma_k(u)| and n is number of vertices)  \Longleftrightarrow  \displaystyle{ \frac{\sum_{u\in V}n/(n-k_d(u))} {\sum_{u\in V}k_d(u)/(n-k_d(u))}= \sum_{i=0}^d \frac{\pi_0^2}{m(\lambda_i)\pi_i^2} } (where \pi_h=\prod\limits_{i=0\atop i\not=h}^d (\lambda_h-\lambda_i) and k_d(u)=|\Gamma_d(u)|). Largest part of main results on which I would like to bring attention, can be found in \cite{MAFiolACDRG}, \cite{SMiklavic}, \cite{MAFiolACOBDRG} and \cite{M. Camara...}.

A connected graph Γ is called nicely distance–balanced, whenever there exists a positive integer γ = γ(Γ), such that for any two adjacent vertices u, v of Γ there are exactly γ vertices of Γ which are closer to u than to v, and exactly γ vertices of Γ which are closer to v than to u. Let d denote the diameter of Γ. It is known that d ≤ γ, and that nicely distance-balanced graphs with γ = d are precisely complete graphs and cycles of length 2d or 2d+1. In this paper we classify regular nicely distance-balanced graphs with γ = d+ 1. Mathematics Subject Classifications: 05C12; 05C75.

Through this thesis we introduce distance-regular graphs, and present some of their characterizations which depend on information retrieved from their adjacency matrix, principal idempotent matrices, predistance polynomials and spectrum. Let $\Gamma$ be a finite simple connected graph. In Chapter I we present some basic results from Algebraic graph theory: we prove Perron-Frobenius theorem, we show how to compute the number of walks of a given length between two vertices of $\Gamma$, how to compute the total number of (rooted) closed walks of a given length of $\Gamma$, we introduce adjaceny matrix $\A$ of $\Gamma$, principal idempotent matrices $\E_i$ of $\Gamma$ and introduce adjacent (Bose-Mesner) algebra of $\Gamma$ and Hoffman polynomial of $\Gamma$. All of these results are needed in Chapters II and III. In Chapter II we define distance-regular graphs, show some examples of these graph, introduce distance-$i$ matrix $\A_i$, $i=0,1,...,D$ (where $D$ is the diameter of graph $\G...

Let \G denote a bipartite distance-regular graph with diameter D \ge 4 and valency k \ge 3. Let X denote the vertex set of \G, and let A denote the adjacency matrix of \G. For x \in X and for 0 \le i \le D, let \G_i(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter \Delta_2 in terms of the intersection numbers by \Delta_2 = (k-2)(c_3-1)-(c_2-1)p^2_{22}. It is known that \Delta_2 = 0 implies that D \le 5 or c_2 \in \{1,2\}. For x \in X let T=T(x) denote the subalgebra of \MX generated by A, \Es_0, \Es_1, \ldots, \Es_D, where for 0 \le i \le D, \Es_i represents the projection onto the ith subconstituent of \G with respect to x. We refer to T as the {\em Terwilliger algebra} of \G with respect to x. By the {\em endpoint} of an irreducible T-module W we mean min\{i | \Es_iW \ne 0\}. We find the structure of irreducible T-modules of endpoint 2 for graphs \G which have the property that for 2\le i\le D-1, there exist complex scalars \alpha_i, \beta_i such that for all x, y, z \in X with \partial(x, y) = 2, \: \partial(x, z) = i, \: \partial(y, z) = i, we have \alpha_i + \beta_i |\G_1(x) \cap \G_1(y) \cap \G_{i-1}(z)| = |\G_{i-1}(x) \cap \G_{i-1}(y) \cap \G_1(z)|, in case when \Delta_2=0 and c_2=2. The case when \Delta_2=0 and c_2=1 is already studied by M. S. MacLean et al. \cite{MacLean_Miklavic_Penjic}. We show that if \G is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module, and we give the action of A on this basis. Free access to this article is valid for 50 days, until January 05, 2017. https://authors.elsevier.com/a/1U3G1,H-c7N0c

Let \Gamma denote a bipartite Q-polynomial distance-regular graph with diameter D \ge 4, valency k \ge 3 and intersection number c_2 \le 2. We show that \Gamma is either the D-dimensional hypercube, or the antipodal quotient of the 2D-dimensional hypercube, or D=5. Paper is available on http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p53