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We consider the notion of χ -completeness of a locally finite graph and we extend this notion to the weighted magnetic graph. Moreover, we establish a link between the magnetic adjacency matrix on line graph and the magnetic discrete Laplacian on 1-forms. [ABSTRACT FROM AUTHOR]

After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene's stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of C40, C60 and C80. This paper suggests a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations (Sure et al. in Phys Chem Chem Phys 19:14296–14305, 2017). We apply our method to classify all isomers of C60 and give an example of two different cospectral isomers of C44. Calculations are done with our own Python scripts available at (Bille et al. in Fullerene database and classification software, https://www.uni-ulm.de/mawi/mawi-stochastik/forschung/fullerene-database/, 2020). The only input for our algorithm is the vector of positions of pentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software package Fullerene (Schwerdtfeger et al. in J Comput Chem 34:1508–1526, 2013). [ABSTRACT FROM AUTHOR]

In the current paper, we study the discrete Laplacian acting on 2-forms which was introduced and investigated by Chebbi (Potential Anal 49(2):331–358, 2018). We establish a new criterion of essential self-adjointness using the Nelson lemma. Moreover, we give an upper bound on the infimum of the essential spectrum. Furthermore, we establish a link between the adjacency matrix and the discrete Laplacian on 2-forms. [ABSTRACT FROM AUTHOR]

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $$-2$$ , or 0, and determine which of these graphs are determined by their adjacency spectrum. [ABSTRACT FROM AUTHOR]

The paper deals with the representation of a transportation infrastructure by a planar connected simple graph and aims at studying its features through the analysis of graph properties. All planar and connected graphs with 4 up to 7 edges are analysed and compared to extract the most suitable parameters to investigate some network features. Then, a set of 41 graphs representing some actual underground networks are also analysed. Besides, as a third scenario, the underground network of Milan, along its development in years, is proposed in order to apply the proposed methodology. Many parameters are taken into consideration. Some of them are already discussed in literature, such as the eigenvalues and gaps of adjacency matrix or such as the 'classical' parameters α, β, γ. Others, such as the first two Betti numbers, are new for these applications. In order to overcome the problem of comparing features of graphs with different size, the normalisation of these parameters is considered. Some relationships between Betti numbers, eigenvalues, and classical parameters are also investigated. Results show that the eigenvalues and gaps of the adjacency matrix well represent some features of the graphs while combining them with the Betti numbers, a more significant interpretation can be achieved. Particularly, their normalised values are able to describe the increasing complexity of a graph. [ABSTRACT FROM AUTHOR]

Let $$\fancyscript{B}(n,r)$$ be the set of all bicyclic graphs with $$n$$ vertices and $$r$$ cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in $$\fancyscript{B}(n,r)$$ . [ABSTRACT FROM AUTHOR]

The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds. [ABSTRACT FROM AUTHOR]

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut edges. [ABSTRACT FROM AUTHOR]

We study spectral properties of the Manhattan products of the path graphs and show the concentration of zero-eigenvelues. [ABSTRACT FROM AUTHOR]

This paper proposes an efficient parallel implementation of the Ramalingam algorithm for dynamically updating the single-sink shortest-path subgraph of a directed graph after adding an edge to the graph with the use of a model of associative (content addressable) parallel systems with vertical processing (the STAR machine). An associative version of this algorithm is described as the InsertNewArc procedure whose correctness is proved. We also consider the main advantages of the associative version of the Ramalingam incremental algorithm. [ABSTRACT FROM AUTHOR]

In this paper we present an O( nlog n) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O( n) time. For general (not planar) graphs, vertex capacities do not make the maximum flow problem more difficult, as there is a simple reduction that eliminates vertex capacities. However, this reduction does not preserve the planarity of the graph. The essence of our algorithm is a different reduction that does preserve the planarity and can be implemented in linear time. For the special case of undirected planar graphs, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw. We also apply our technique to obtain a linear-time algorithm to convert a flow to an acyclic flow, and a linear-time algorithm to find a largest set of vertex-disjoint s− t paths, in a directed planar graph. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively. [ABSTRACT FROM AUTHOR]

The Hoffman-Singleton graph is the unique strongly regular graph with parameters (50, 7, 0, 1). A well-known hypothesis states that a distance-regular graph in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph has intersection array {50, 42, 1; 1, 2, 50} or {50, 42, 9; 1, 2, 42}. In the present paper, we prove this hypothesis under the assumption that a distance-regular graph is a Terwilliger graph and the graph diameter is at most 5. [ABSTRACT FROM AUTHOR]

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note, we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium. [ABSTRACT FROM AUTHOR]

Possible orders and fixed-point subgraphs of automorphisms of a generalized octagon of order (2, 4) are found. The vertex-symmetric generalized octagon of order (2, 4) is proved to be classical. [ABSTRACT FROM AUTHOR]

It is proved that any set of finite pairwise nonisomorphic weakly cospectral connected pseudographs, as well as any set of pseudodigraphs satisfying various additional conditions, is finite. [ABSTRACT FROM AUTHOR]