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 {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}

The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development.     

Automorphisms of graphs with intersection arrays {60, 45, 8; 1, 12, 50} and {49, 36, 8; 1, 6, 42}

Automorphisms of distance-regular graphs are considered. It is proved that any graph with the intersection array {60, 45, 8; 1, 12, 50} is not vertex symmetric, and any graph with the intersection array {49, 36, 8; 1, 6, 42} is not edge symmetric.     

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 \(\hat{\Delta }\).     

A constant bound on the number of columns (1,k − 2, 1) in the intersection array of a distance-regular graph

We show that the number of columns (1,k – 2, 1) in the intersection array of distance-regular graphs is at most 20.     

Distance-regular graphs with intersection arrays {52, 35, 16; 1, 4, 28} and {69, 48, 24; 1, 4, 46} do not exist

We prove that the arrays {52, 35, 16; 1, 4, 28} and {69, 48, 24; 1, 4, 46} cannot be realized as the intersection arrays of distance-regular graphs. In the proof we use some inequalities bounding the size of substructures (cliques, cocliques) in a distance-regular graph.     

A Shilla graph with Intersection Array {12, 10, 2; 1, 2, 8} Does not Exist

Mathematical Notes -