Intersection Array Research Articles

On automorphisms of a distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}

Possible prime-order automorphisms and their fixed-point subgraphs are found for a hypothetical distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}. It is shownthat graphs with intersection arrays {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, and {39, 36, 1; 1, 2, 39} are not vertex-symmetric.

On automorphisms of a distance-regular graph with intersection array {204, 175, 48, 1; 1, 12, 175, 204}

A distance-regular graph Γ with intersection array {204, 175, 48, 1; 1, 12, 175, 204} is an AT4-graph, and the antipodal quotient \(\overline \Gamma \) has parameters (800, 204, 28, 60). Automorphisms of these graphs are found. In particular, neither of the two graphs is arc-transitive.

The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results

It is known that, up to isomorphism, there is a unique distance-regular graph Delta with intersection array {32,27;1,12} [equivalently, Delta is the unique strongly regular graph with parameters (105, 32, 4, 12)]. Here we investigate the distance-regular antipodal covers of Delta . We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of Delta (a graph hat{Delta } discovered by the author over 20 years ago), proving that there is a unique distance-regular graph with intersection array {32,27,8,1;1,4,27,32}. In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of Delta , and so no distance-regular graph with intersection array {32,27,6,1;1,6,27,32}. We also show there is no distance-regular antipodal 4-cover of Delta , and so no distance-regular graph with intersection array {32,27,9,1;1,3,27,32}, and that there is no distance-regular antipodal 6-cover of Delta that is a double cover of hat{Delta }.

The Nonexistence of a Distance-Regular Graph with Intersection Array {22,16,5;1,2,20}

We prove that a distance-regular graph with intersection array {22,16,5;1,2,20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local graph. Then we construct a partial linear space from the local graph to display the contradiction.

On automorphisms of distance-regular graphs with intersection arrays {2r + 1, 2r − 2, 1; 1, 2, 2r + 1}

On automorphisms of distance-regular graphs with intersection arrays {2r + 1, 2r − 2, 1; 1, 2, 2r + 1}

On Automorphisms of Distance-Regular Graph with Intersection Array $$\{18,15,9;\,1,1,10\}$$ { 18 , 15 , 9 ; 1 , 1 , 10 }

Recently, Makhnev and Nirova found intersection arrays of distance-regular graphs with \(\lambda =2\) and at most 4096 vertices. In the case of primitive graphs of diameter 3 with \(\mu = 1\) there corresponding arrays are \(\{18,15,9;1,1,10\}\), \(\{33,30,8;1,1,30\}\) or \(\{39,36,4;1,1,36\}\). In this work, possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array \(\{18,15,9;1,1,10\}\) are studied. In particular, graph with intersection array \(\{18,15,9;1,1,10\}\) is not vertex symmetric.

Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3

The study of graphs with the property specified in the title has earlier been reduced to the case of neighborhoods with parameters (35, 18, 9, 9), (36, 21, 12, 12), (40, 27, 18,18), (50, 28, 15, 16), (56, 45, 36, 36), and (64, 27, 10, 12). It is proved that a completely regular graph in which the neighborhoods of vertices are strongly regular graphs with the corresponding parameters is either a strongly regular graph with parameters (76, 35, 18, 14) or (64, 35, 18, 20) or a Taylor graph with intersection array {35, 16, 1; 1, 16, 35} (this is the main theorem). As a consequence, a classification of distance regular graphs in which the neighborhoods of vertices are strongly regular graphs with a non principal eigenvalue r, 2 < r ≤ 3, is obtained.

On extensions of small strongly regular graphs with eigenvalue 3

Previously, the authors found the parameters of exceptional strongly regular graphs with eigenvalue 3 (theorem). It is proved that, if Γ is a distance-regular graph whose local subgraphs are strongly regular with parameters listed in items (3)–(10) of the theorem, then Γ is a strongly regular graph, Taylor graph, or a graph with intersection array {100, 66, 1; 1, 33, 100}.

On automorphisms of a distance-regular graph with intersection array {35, 32, 1; 1, 4, 35}

Prime divisors of orders of automorphisms and their fixed-point subgraphs are studied for a hypothetical distance-regular graph with intersection array {35, 32, 1; 1, 4, 35}. It is shown that there are no arc-transitive distance-regular graphs with intersection array {35, 32, 1; 1, 4, 35}.

On automorphisms of a distance-regular graph with intersection array {15, 12, 1; 1, 4, 15}

On automorphisms of a distance-regular graph with intersection array {15, 12, 1; 1, 4, 15}

On automorphisms of a distance-regular graph with intersection array {33, 30, 15; 1, 2, 15}

On automorphisms of a distance-regular graph with intersection array {33, 30, 15; 1, 2, 15}

Automorphisms of a graph with intersection array {115, 96, 30, 1; 1, 10, 96, 115}

Automorphisms of a graph with intersection array {115, 96, 30, 1; 1, 10, 96, 115}

Automorphisms of a graph with intersection array {99, 84, 1; 1, 14, 99}

Automorphisms of a graph with intersection array {99, 84, 1; 1, 14, 99}

The Hosoya polynomial of distance-regular graphs

In this note we obtain an explicit formula for the Hosoya polynomial of any distance-regular graph in terms of its intersection array. As a consequence, we obtain a very simple formula for the Hosoya polynomial of any strongly regular graph.

On strongly regular graphs with eigenvalue μ and their extensions

Let M be a class of strongly regular graphs for which μ is a nonprincipal eigenvalue. Note that the neighborhood of any vertex of an AT4-graph lies in M. We describe parameters of graphs from M and find intersection arrays of AT4-graphs in which neighborhoods of vertices lie in chosen subclasses from M. In particular, an AT4-graph in which neighborhoods of vertices do not contain triangles is the Conway-Smith graph with parameters (p, q, r) = (1, 2, 3) or the first Soicher graph with parameters (p, q, r) = (2, 4, 3).

Step-size sequence design for finite-time average consensus in secure wireless sensor networks

This paper concerns the study of average consensus in wireless sensor networks with aim of providing a way to reach consensus in a finite number of steps. In particular, we investigate the design of consensus protocols when, for security reasons for instance, the underlying graph is constrained to be strongly regular or distance regular. The proposed design method is based on parameters of the intersection array characterizing the underlying graph. With this protocol, at execution time, average consensus is achieved in a number of steps equal to the diameter of the graph, i.e. the smallest possible number of steps to achieve consensus.

Arc transitive distance regular covers of cliques

Abstract It is consider edge-symmetric distance-regular graph Γ with intersection array { k , ( r − 1 ) μ , 1 ; 1 , μ , k } . Let G = Aut ( Γ ) . Then G acts two-transitively on the set Σ of antipodal classes (fibres) of Γ. This graphs are classified in the next cases: 1) r = 2 or r = k , 2) λ = μ , 3) G contains abelian normal subgroup N regular on Σ (affine case).

On geometric distance-regular graphs with diameter three

In this paper we study distance-regular graphs with intersection array (1){(t+1)s,ts,(t−1)(s+1−ψ);1,2,(t+1)ψ} where s,t,ψ are integers satisfying t≥2 and 1≤ψ≤s. Geometric distance-regular graphs with diameter three and c2=2 have such an intersection array. We first show that if a distance-regular graph with intersection array (1) exists, then s is bounded above by a function in t. Using this we show that for a fixed integer t≥2, there are only finitely many distance-regular graphs of order (s,t) with smallest eigenvalue −t−1, diameter D=3 and intersection number c2=2 except for Hamming graphs with diameter three. Moreover, we will show that if a distance-regular graph with intersection array (1) for t=2 exists then (s,ψ)=(15,9). As Gavrilyuk and Makhnev (2013) [9] proved that the case (s,ψ)=(15,9) does not exist, this enables us to finish the classification of geometric distance-regular graphs with smallest eigenvalue −3, diameter D≥3 and c2≥2 which was started by the first author (Bang, 2013) [1].

The Spectral Excess Theorem for Distance-Biregular Graphs.

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph $\Gamma$ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the spectral excess theorem for bipartite distance-biregular graphs.

On graphs whose local subgraphs are strongly regular with parameters (115, 18, 1, 3)

We consider undirected graphs without loops or multiple edges. If a and b are vertices in a graph Γ, then d(a, b) denotes the distance between a and b, and Γi(a) denotes the subgraph of Γ induced by the set of vertices of Γ that are a distance of i away from a. The subgraph Γ1(a) is called the neighborhood of a and is denoted by (a). Γ is called a regular graph of degree k if (a) contains precisely k vertices for any vertex a in Γ. Γ is called an edgeregular graph with parameters ( v, k, λ) if Γ is a regular graph of degree k on v vertices and each edge in Γ lies in λ triangles. Γ is said to be an amply regular graph with parameters (v, k, λ, μ) if Γ is an edgereg� ular graph with the corresponding parameters and the subgraph (a) ∩ (b) contains μ vertices in the case of d(a, b) = 2. An amply regular graph of diameter 2 is called a strongly regular graph. A graph Γ of diameter d is said to be antipodal if the relation of coincidence or being a distance of d apart on its vertex set is an equivalence relation. An antipo� dal quotient Γ' is a graph whose vertices are the antip� odal classes of Γ and two classes are adjacent if a vertex of one class is adjacent to a vertex of the other class. An antipodal graph Γ is called an rcovering (of its antip� odal quotient) if each of its antipodal classes contains precisely r vertices. If vertices u and w are separated by a distance of i in Γ, then bi(u, w) (ci(u, w)) denotes the number of vertices in the intersection of Γi +1 (u) (Γi -1 (u)) with (w). A graph Γ of diameter d is called a distanceregular graph with the intersection array {b 0 , b 1 , …, b d -1 ; c 1 , c2, …, cd} if the values bi(u, w) and ci(u, w) are indepen� dent of the choice of the vertices u and w separated by a distance of i. Let ai = k - bi - ci.