Intersection Array Research Articles

Efficient data acquisition and reconstruction for air-coupled ultrasonic robotic NDE

Non-destructive evaluation of complex parts using surface scanning techniques, such as ultrasonic testing and eddy current testing, requires complex manipulation of such sensors to ensure quantitative results. A robotic arm may function as a complex manipulator for surface scanning, controlling the position and tilt between the probe and specimen’s surface. To ensure accuracy in probe manipulation, accurate geometric information of the specimen is required. This article explores a methodology that uses structured light for physical-to-virtual reconstruction, providing submillimeter scale and accurate surface geometries. Reconstruction aids in path planning through a novel ray-triangle intersection array algorithm, establishing movements for the NDE probe to orient itself on the specimen at a constant probe to specimen surface distance, or lift-off. The proposed technique is demonstrated and validated through experimental air-coupled ultrasonic inspection of automotive CFRP composite samples with simulated flaws such as interlaminar delamination. The proposed method employs guided waves and a pitch-catch configuration of air-coupled ultrasonic probes, enabling single-side access scans. A Fanuc 100ib robot arm was used to manipulate the ultrasonic probes along a sample reconstructed with a CR-Scan 01 structured light sensor. The probes were excited at 200khz from a SonoAir system, while also recovering defect vs background information synchronized with the probe’s orientation. Additionally, a framework for potential automation is proposed, with further details to be explored in future works.

Об автоморфизмах графов с массивами пересечений {44,40,12;1,5,33} и {48,35,9;1,7,40}

Distance-regular graph Γ of diameter 3 with strongly regular graphs Γ2 and Γ3 has intersection array {r(c2+1)+a3, r c2, a3 + 1; 1, c2, r(c2 + 1)} (M.S. Nirova). For distance-regular graph Γ of diameter 3 and degree 44 there are exactly 7 feasible intersection arrays. For each of them graph Γ3 is strongly regular. For intersection array {44, 30, 5; 1, 3, 40} we have a3 = 4, c2 = 3, r = 10, Γ2 has parameters (540,440,358,360) and Γ3 has parameters (540,55,10,5). Graph does not exist (Koolen-Park). For intersection array {44, 35, 3; 1, 5, 42} graph Γ3 has parameters (375,22,5,1) and does not exist (its neighbourhood of vertex is the union of isolated 6-cliques). In this paper it is found futomorphisms of graphs with intersection arrays {44,40,12; 1,5,33 and 48,35,9; 1,7,40}.

On distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph

Abstract There exist well-known distance-regular graphs Γ of diameter 3 for which Γ 3 is a triangle-free graph. An example is given by the Johnson graph J (8, 3) with the intersection array {15, 8, 3;1, 4, 9}. The paper is concerned with the problem of the existence of distance-regular graphs Γ with the intersection arrays {78, 50, 9;1, 15, 60} and {174, 110, 18;1, 30, 132} for which Γ 3 is a triangle-free graph.

Q-полиномиальный граф с массивом пересечений {60, 45, 8; 1, 12, 50} не существует

When studying amply regular graphs Γ of diameter d, in which for some vertex a the pair (Γd(a), Γd-1(a)) is a 2-scheme, it is proved that the subgraph induced by the set of points is a clique, coclique, or strongly regular graph. For a graph of diameter 3, it is established that this construction is a 2-scheme for any vertex a if and only if the graph is distance-regular and for any vertex a the subgraph Γ3(a) is a clique, coclique, or strongly regular graph (A.L. Gavrilyuk, A.A. Makhnev). An interesting question is whether there is a distance-regular graph with intersection array {60,45,8;1,12,50} such that Γ3(a) can be a 6×6-lattice and the pair (Γ3(a), Γ2(a)) will be a 2-scheme. In the paper of I.N. Belousov and A.A. Makhnev (2018) published a proof of the non-existence of the abovementioned graph, that contained errors. In this paper, we give a correct proof of this result.

ON DISTANCE–REGULAR GRAPHS OF DIAMETER 3 WITH EIGENVALUE \(\theta=1\)

For a distance-regular graph \(\Gamma\) of diameter 3, the graph \(\Gamma_i\) can be strongly regular for \(i=2\) or 3. J.Kulen and co-authors found the parameters of a strongly regular graph \(\Gamma_2\) given the intersection array of the graph \(\Gamma\) (independently, the parameters were found by A.A. Makhnev and D.V.Paduchikh). In this case, \(\Gamma\) has an eigenvalue \(a_2-c_3\). In this paper, we study graphs \(\Gamma\) with strongly regular graph \(\Gamma_2\) and eigenvalue \(\theta=1\). In particular, we prove that, for a \(Q\)-polynomial graph from a series of graphs with intersection arrays \(\{2c_3+a_1+1,2c_3,c_3+a_1-c_2;1,c_2,c_3\}\), the equality \(c_3=4 (t^2+t)/(4t+4-c_2^2)\) holds. Moreover, for \(t\le 100000\), there is a unique feasible intersection array \(\{9,6,3;1,2,3\}\) corresponding to the Hamming (or Doob) graph \(H(3,4)\). In addition, we found parametrizations of intersection arrays of graphs with \(\theta_2=1\) and \(\theta_3=a_2-c_3\).

The average search probabilities of discrete-time quantum walks

We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertex-deleted subgraph. In particular, for any family of the average search probability approaches 1/4 as the valency goes to infinity. We also present a more relaxed criterion, in terms of the intersection array, for this limit to be approached by distance regular graphs.

Distance-regular graphs admitting a perfect 1-code

In this paper, we study the problem that which of distance-regular graphs admit a perfect 1-code. Among other results, we characterize distance-regular line graphs which admit a perfect 1-code. Moreover, we characterize all known distance-regular graphs with small valency at most 4, the distance-regular graphs with known putative intersection arrays for valency 5, and all distance-regular graphs with girth 3 and valency 6 or 7 which admit a perfect 1-code.

Некоторые графы Шилла с b = 5 не существуют

A Shilla graph is a distance-regular graph of diameter 3 that has a second eigenvalue equal to a = a3 . Koolen and Park found admissible arrays of intersections of the Shill graphs with b = 3 (there were 12 of them). Belousov I.N. found feasible intersection arrays of the Shilla graphs with b = 4 (there were 50 of them) and b = 5 (there were 82 of them). It is proved in the paper that distance-regular Schill graphs with b = 5 and intersection arrays {305,248,62;1,2,244}, {315,256,64;1,2,252}, {345,280,64;1,4,276},{615,496,124;1,4,492}, {815,656,164;1,2,652}, {855,688,172;1,4,684}, {855,688,170;1,5,684}, {910,732,180;1,10, 728}, {1000,804,201;1,3,800}, {1045,840,210;1,6,836}, {1055,848,212;1,4,844}, {1080,868,215;1,5,864}, {1155,928,232;1,2,924}, {1185,952,245;1,5,948}, {1235,992,248;1,8,988},{1535,1232,308;1,8,1228}, {1560,1252,310;1,10,1248}, {1615,1296,324;1,12,1292}, {1665,1336,334;1,2,1332} do not exist.

Transparent photodetectors based on polyoxometalate modified electrospun ZnO homojunction nanowire intersection arrays

We use electrospinning to modify ZnO NWs with POMs. The composite exhibits the features of a p-type semiconductor and matches the Eg of ZnO. The PDs prepared by POMs@ZnO NWs can work under zero bias conditions and show good light responsivity.

SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)

A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.

On the distance spectrum of minimal cages and associated distance biregular graphs

A (k,g)-cage is a k-regular simple graph of girth g with minimum possible number of vertices. In this paper, (k,g)-cages which are Moore graphs are referred as minimal (k,g)-cages. A simple connected graph is called distance regular (DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular (DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal (k,g)-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal (k,g)-cages we give a formula for distance spectral radius in terms of k and g, and also determine polynomials of degree ⌊g2⌋, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal (k,g)-cage of diameter d has d+1 distinct distance eigenvalues, and this partially answers a problem posed in [1]. We prove that every DBR graph is a 2-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal (k,g)-cages. Finally we determine the full distance spectrum of subdivision of some minimal (k,g)-cages.

Distance-regular Terwilliger graphs with intersection arrays {50,42,1;1,2,50} and {50,42,9;1,2,42} do not exist

Distance-regular Terwilliger graphs with intersection arrays {50,42,1;1,2,50} and {50,42,9;1,2,42} do not exist

On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

On nonexistence of distance regular graphs with the intersection array ${53,40,28,16;1,4,10,28}$

The prime spectrum of an automorphism group of an 𝐴𝑇4(𝑝,𝑝+2,𝑟)-graph

The present paper is devoted classification of A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graphs. There is a unique A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph with p = 2 p=2 , namely, the distance-transitive Soicher graph with intersection array { 56 , 45 , 16 , 1 ; 1 , 8 , 45 , 56 } \{56, 45, 16, 1;1, 8, 45, 56\} , whose local graphs are isomorphic to the Gewirtz graph. The existence of an A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph with p > 2 {p>2} remains an open question. It is known that the local graphs of each A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph are strongly regular with parameters ( ( p + 2 ) ( p 2 + 4 p + 2 ) , p ( p + 3 ) , p − 2 , p ) \big ((p+2)(p^2+4p+2),p(p+3),p-2,p\big ) . In this paper, an upper bound is found for the prime spectrum of the automorphism group of a strongly regular graph with such parameters, and also some restrictions obtained for the prime spectrum and the structure of the automorphism group of an A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph in the case when 2 > p 2>p is a prime power. As a corollary, it is shown that there are no arc-transitive A T ( p , p + 2 , r ) \mathrm {AT}(p,p+2,r) -graphs with p ∈ { 11 , 17 , 27 } p\in \{11,17,27\} .

On the Nonexistence of Distance-Regular Graphs with Intersection Array $$ \{ 53,40,28,16; 1,4,10,28\}$$

On the Nonexistence of Distance-Regular Graphs with Intersection Array $$ \{ 53,40,28,16; 1,4,10,28\}$$

Automorphisms of a Distance Regular Graph with Intersection Array $\{21,18,12,4;1,1,6,21\}$

А. А. Махнев и М. С. Нирова нашли массивы пересечений дистанционно регулярных графов с $\lambda=2$ и числом вершин, не большим 4096. Для графов диаметра $4$ наиболее интересен массив $\{21,18,12,4;1,1,6,21\}$ из этого списка. В данной работе найдены возможные порядки и подграфы неподвижных точек автоморфизмов дистанционно регулярного графа с массивом пересечений $\{21,18,12,4;1,1,6,21\}$. Библиография: 5 названий.

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

Automorphisms of a Distance Regular Graph with Intersection Array $$\{21,18,12,4;1,1,6,21\}$$

DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST

In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.

The largest Moore graph and a distance regular graph with intersection array ${55,54,2;1,1,54}$

The largest Moore graph and a distance regular graph with intersection array ${55,54,2;1,1,54}$

The Largest Moore Graph and a Distance-Regular Graph with Intersection Array {55, 54, 2; 1, 1, 54}

We point out possible automorphisms of a distance-regular graph Γ with intersection array {55, 54, 2; 1, 1, 54} and spectrum 551, 71617,−1110,−81408.