On the maximal distance spectral radius in a class of bicyclic graphs
01 Dec 2012-Discrete Mathematics, Algorithms and Applications (World Scientific Publishing Company)-Vol. 04, Iss: 04, pp 1250061
TL;DR: The distance spectral radius of bicyclic graphs in $\mathcal{B}_{n}$ is studied, and the graph with the largestdistance spectral radius is determined.
Abstract: Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let Pp+1 = x1x2⋯xp+1, Pt+1 = y1y2⋯yt+1 and Pq+1 = z1z2⋯zq+1 be three vertex-disjoint paths. Identifying the initial vertices as u0 and the terminal vertices as v0, the resultant graph, denoted by θ(p; t; q), is called a θ-graph. Let $\mathcal{B}_{n}$ be the class of all bicyclic graphs on n vertices, which contain a θ-graph as an induced subgraph. In this paper, we study the distance spectral radius of bicyclic graphs in $\mathcal{B}_{n}$, and determine the graph with the largest distance spectral radius.
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TL;DR: The present paper reports on the results related to the distance matrix of a graph and its spectral properties.
233 citations
01 Nov 2013
TL;DR: In this paper, the spectral properties of the distance matrix of a connected graph and its spectral properties were investigated and the authors reported the results related to the distance matrices of a graph and their spectral properties.
Abstract: In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.
212 citations
TL;DR: In this paper, the unique graphs with minimum and second-minimum distance (distance signless Laplacian, respectively) spectral radii were determined among bicyclic graphs with fixed number of vertices.
53 citations
TL;DR: In this article, the unique graph with maximal distance spectral radius in the class of graphs without a pendent vertex was determined, where the spectral radius of the graph is the same as that of the vertex.
19 citations
TL;DR: This work determines the unique graphs with minimum and second minimum distance spread in the class of cacti and in theclass of bicyclic graphs, respectively.
4 citations
Cites background from "On the maximal distance spectral ra..."
...Paul [10] determined the unique graph with maximum distance spectral radius among bicyclic graphs containing three internally vertex disjoint paths with common end vertices....
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References
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Book•
01 Jan 1995
TL;DR: The Spectrum and the Group of Automorphisms as discussed by the authors have been used extensively in Graph Spectra Techniques in Graph Theory and Combinatory Applications in Chemistry an Physics. But they have not yet been applied to Graph Spectral Biblgraphy.
Abstract: Introduction. Basic Concepts of the Spectrum of a Graph. Operations on Graphs and the Resulting Spectra. Relations Between Spectral and Structural Properties of Graphs. The Divisor of a Graph. The Spectrum and the Group of Automorphisms. Characterization of Graphs by Means of Spectra. Spectra Techniques in Graph Theory and Combinatories. Applications in Chemistry an Physics. Some Additional Results. Appendix. Tables of Graph Spectra Biblgraphy. Index of Symbols. Index of Names. Subject Index.
2,119 citations
TL;DR: The addressing scheme proposed will be applied primarily to local loops where the mutual interconnections may be quite varied and if a certain amount of hierarchical structure is introduced into the regional and national loop structure, as suggested by J. R. Pierce1 it is possible to achieve addressings which are both compact and quite efficient.
Abstract: The methods used to perform the switching functions of the Bell System have been developed under the fundamental assumption that the holding time of the completed call is long compared to the time needed to set up the call. In considering certain forms of communication with and among computers the possibility arises that a message, with its destination at its head might thread its way through a communication network without awaiting the physical realization of a complete dedicated path before beginning on its journey. One such scheme has been proposed by J. R. Pierce and may be called “loop switching.” We imagine subscribers, perhaps best thought of as computer terminals or other data generating devices, on one-way loops. These “local” loops are connected by various switching points to one another as well as to other “regional” loops which are in turn connected to one another as well as to a “national” loop. If a message from one loop is destined for a subscriber on another loop it proceeds around the originating loop to a suitable switching point where it may choose to enter a different loop, this process continuing until the message reaches its destination. The question naturally comes up, how the message is to know which sequence of loops to follow. It would be desirable for the equipment at each junction to be able to apply a simple test to the destination addressing the loops which has several attractive features: (i) It permits an extremely simple routing strategy to be used by the messages in reaching their destmations. (ii) By using this strategy, a message will always take the shortest possible path between any two local loops in the same region. {iii) The method of addressing applies to any collection of loops, no matter hoio complex their interconnections. The addressing scheme we propose will be applied primarily to local loops where the mutual interconnections may be quite varied. If a certain amount of hierarchical structure is introduced into the regional and national loop structure, as suggested by J. R. Pierce1 it is possible to achieve addressings which are both compact and quite efficient.
477 citations
"On the maximal distance spectral ra..." refers background in this paper
...The distance matrix of a graph has come up in several different areas, including communication network design [5], graph embedding theory [2, 4] and network flow algorithms [3]....
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TL;DR: It is shown that $O(N^{5/2} )$ comparisons and additions suffice to solve the all-pairs shortest path problem for directed graphs on N vertices with nonnegative edge weights.
Abstract: It is shown that $O(N^{5/2} )$ comparisons and additions suffice to solve the all-pairs shortest path problem for directed graphs on N vertices with nonnegative edge weights. In conjunction with preprocessing, this result is exploited to produce an $o(N^3 )$ algorithm for solving the shortest path problem.
280 citations
"On the maximal distance spectral ra..." refers background in this paper
...The distance matrix of a graph has come up in several different areas, including communication network design [5], graph embedding theory [2, 4] and network flow algorithms [3]....
[...]
TL;DR: In this article, the authors describe exactly how the coefficients δK(T) depend on the structure of a tree T. In contrast to the corresponding problem for the adjacency matrix of T, the results here are surprisingly difficult, requiring the use of a number of interesting auxiliary results.
223 citations
"On the maximal distance spectral ra..." refers background in this paper
...The distance matrix of a graph has come up in several different areas, including communication network design [5], graph embedding theory [2, 4] and network flow algorithms [3]....
[...]
TL;DR: Let T be a tree with line graph T*.
Abstract: Let T be a tree with line graph T*. Define K = 21 + A(T*), where A denotes the adjacency matrix. Then the eigenvalues of -2 K-’ interlace the eigenvalues of the distance matrix D. This permits numerous results about the spectrum of K to be transcribed for the less tractable D.
104 citations
"On the maximal distance spectral ra..." refers background in this paper
...For related results, one may refer to [6, 7, 10] and the references therein....
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