On strongly regular graphs with m2 = qm3 and m3 = qm2

2010 Mathematics Subject Classification: 05C50.We say that a regular graph G of order n and degree r і 1 (which is not the complete graph) is strongly regular if there exist non-negative integers t and q such that |SiЗSj| = t for any two adjacent vertices i and j, and |SiЗSj| = q for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let l1 = r, l2 and l3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, l2 and l3, respectively. We here describe the parameters n, r, t and q for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 2, 3, 4

On strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 7/2, 7/3, 7/4, 7/5, 7/6

We say that a regular graph G of order n and degree r ? 1 (which is not the complete graph) is strongly regular if there exist non-negative integers ? and ? such that |Si ? Sj| = ? for any two adjacent vertices i and j, and |Si ? Sj| = ? for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let ?1 = r, ?2 and ?3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, ?2 and ?3, respectively. We here describe the parameters n, r, ? and ? for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 7/2, 7/3, 7/4, 7/5, 7/6.</jats:p

On strongly regular graphs with m2 = qm3 and m3 = qm2 where q ∈ Q

We say that a regular graph G of order n and degree r ? 1 (which is not the complete graph) is strongly regular if there exist non-negative integers ? and ? such that |Si ? Sj | = ? for any two adjacent vertices i and j, and |Si ? Sj | = ? for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let ?1 = r, ?2 and ?3 be the distinct eigenvalues of a connected strongly regular graph. Let m1 = 1, m2 and m3 denote the multiplicity of r, ?2 and ?3, respectively. We here describe the parameters n, r, ? and ? for strongly regular graphs with m2 = qm3 and m3 = qm2 for q = 3/2, 4/3, 5/2, 5/3, 5/4, 6/5.</jats:p

Choosing the exponent in the definition of the connectivity index

Let du denote the degree of the vertex u of a molecular graph G. Then the connectivity index of G is defined as C (l) = G (l; C) = S (dudu)l, where the summation goes over all pairs of adjacent vertices. The exponent l is usually chosen to be equal to 1/2, but other options were considered as well, especially l = 1. We show that whereas C(1/2) is a suitable measure of branching of the carbon-atom skeleton of organic molecules, and thus applicable as a topological index for modeling physico-chemical properties of the respective compounds, this is not the case with C(1). The value of l is established, beyond which C(l) fails to correctly reflect molecular branching

Choosing the exponent in the definition of the connectivity index

Let ?v denote the degree of the vertex v of a molecular graph G. Then the connectivity index of G is defined as C (?) = G (?,C) = ? (?u?v)?, where the summation goes over all pairs of adjacent vertices. The exponent ? is usually chosen to be equal to -1/2, but other options were considered as well, especially ?=-1. We show that whereas C(-1/2) is a suitable measure of branching of the carbon-atom skeleton of organic molecules, and thus applicable as a topological index for modeling physico-chemical properties of the respective compounds, this is not the case with C(-1). The value of ? is established, beyond which C(?) fails to correctly reflect molecular branching.</jats:p

The high-energy band in the photoelectron spectrum of alkanes and its dependence on molecular structure

In the model for the ionization energies of the C2s-electrons in saturated hydrocarbons, put forward by Heilbronner et al., the energy levels are calculated as eigenvalues of the line graph of the hydrogen-filled molecular graph. It is now shown that in the case of alkanes, these energy levels are related to the Laplacian eigenvalues of the molecular graph. A few rules are formulated, relating these ionization energies with molecular structure.</jats:p

The high-energy band in the photoelectron spectrum of alkanes and its dependence on molecular structure

In the model for the ionization energies of the C2s-electrons in saturated hydrocarbons, put forward by Heilbronner et al., the energy levels are calculated as eigenvalues of the line graph of the hydrogen-filled molecular graph. It is now shown that in the case of alkanes, these energy levels are related to the Laplacian eigenvalues of the molecular graph. A few rules are formulated, relating these ionization energies with molecular structure

Comparing Cross-classified Mixed Effects and Bayesian Structural Equations Modeling for Stimulus Sampling Designs: A Simulation Study

Researchers examining a wide range of psychological phenomena, including interpersonal perception, attitude formation, and stereotype activation, apply stimulus sampling designs (SSD). The standard SSD study requires participant raters to provide evaluations of a series of target stimuli (e.g., photographs, media clips, vignettes), and the constituent responses are simultaneously nested within participants and stimuli, yielding a cross-classified data structure. Prior methodological work has illustrated the application of both cross-classified mixed effects, and cross-classified structural equation modeling to accommodate the corresponding dependency structure. Despite their widespread application, little is known about how sample size for both participants and stimuli is associated with inferential power and coverage in SSDs. Even less is known about the feasibility of random slopes, or whether the corresponding frequentist (maximum likelihood) and Bayesian (MCMC) estimators differ in accuracy or efficiency under the design conditions typically observed in SSD studies. We conducted a Monte Carlo simulation study to better understand parameter bias, statistical power, and confidence or credible interval coverage, as a function of the number of participant raters and target stimuli, effect size, as well as the presence of random slopes, and modeling framework. Findings revealed that the CC-MEM and CC-BSEM approaches provided very similar point estimates and statistical inference conclusions for the fixed slopes model, though a number of discrepancies arose when random slopes were estimated. Recommendations for future research are provided

Exponent-dependent properties of the connectivity index

457-461The connectivity index is defined as C(λ) =Σ(δuδv)λ, where δv is the degree of the vertex v of the respective molecular graph, and where the summation embraces all pairs of adjacent vertices. The exponent λ is usually chosen to be equal to -0.5 but other options have been considered as well, especially C(-1). We show that whereas C(-0.5) correctly reflects the extent of branching of the carbon-atom skeleton of organic molecules, and is thus a suitable topological index for modelling physico-chemical properties of the respective compounds, this is not the case when the exponent λ assumes larger negative values, in particular when λ= -1.The value of λ is established beyond which C(λ) fails to be a measure of molecular branching