The main eigenvalues and number of walks in self-complementary graphs

Farrugia, Alexander; Sciriha, Irene

doi:10.1080/03081087.2013.825959

Cited by 5 publications

(7 citation statements)

References 7 publications

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“…The following result is immediate. For future reference, we denote the four pairs of controllable graphs on six vertices by (CP 6,1 , CP 6,1 ), (CP 6,2 , CP 6,2 ), (CP 6,3 , CP 6,3 ) and (CP 6,4 , CP 6,4 ). These graphs can be identified from [2] as being the graphs with identification numbers (98, 46), (87, 67), (85, 60) and (77, 59) respectively.…”

Section: Theorem 48 Every Controllable Graph G On At Least Six Vertmentioning

confidence: 99%

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On strongly asymmetric and controllable primitive graphs

Farrugia

2016

Discrete Applied Mathematics

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Section: Theorem 48 Every Controllable Graph G On At Least Six Vertmentioning

confidence: 99%

“…In [6], it was proved that the only self-complementary controllable graph is K 1 . We can rewrite this result as in Theorem 4.5.…”

Section: Theorem 43 Any Controllable Graph Is One Of the Following mentioning

confidence: 99%

On strongly asymmetric and controllable primitive graphs

Farrugia

2016

Discrete Applied Mathematics

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“…The unique vectors z (1) , z (2) ,…, z ( p ) of Lemma 4 are referred to as the main eigenvectors of G [ 5 , 6 ].…”

Section: Local Centralitiesmentioning

confidence: 99%

Balanced Centrality of Networks

Debono

Lauri

Sciriha

2014

International Scholarly Research Notices

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There is an age-old question in all branches of network analysis. What makes an actor in a network important, courted, or sought? Both Crossley and Bonacich contend that rather than its intrinsic wealth or value, an actor's status lies in the structures of its interactions with other actors. Since pairwise relation data in a network can be stored in a two-dimensional array or matrix, graph theory and linear algebra lend themselves as great tools to gauge the centrality (interpreted as importance, power, or popularity, depending on the purpose of the network) of each actor. We express known and new centralities in terms of only two matrices associated with the network. We show that derivations of these expressions can be handled exclusively through the main eigenvectors (not orthogonal to the all-one vector) associated with the adjacency matrix. We also propose a centrality vector (SWIPD) which is a linear combination of the square, walk, power, and degree centrality vectors with weightings of the various centralities depending on the purpose of the network. By comparing actors' scores for various weightings, a clear understanding of which actors are most central is obtained. Moreover, for threshold networks, the (SWIPD) measure turns out to be independent of the weightings.

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“…, v r ) such that v 0 = x, v r = y and v i−1 is adjacent to v i for all 1 ≤ i ≤ r. If x = y then the walk is called a closed walk of length r at vertex x. The number of walks in a graph is often necessary in, for instance, network analysis, epidemiology (requiring slow diffusion of viruses) and network design (aiming for fast data propagation) [3]. Also walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry [6].…”

Section: Introductionmentioning

confidence: 99%

On the Walks on Cayley Graphs‎

Arezoomand¹

2019

FU Math Inform

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‎Let $G$ be a group and $S$ be an inverse-closed subset of $G$ not contining of the identity element of $G$‎. ‎The Cayley graph‎‎of $G$ with respect to $S$‎, ‎$\Cay(G,S)$‎, ‎is a graph with vertex set $G$ and edge set $\{\{g,sg\}\mid g\in G,s\in S\}$‎.‎In this paper‎, ‎we compute the number of walks of any length between two arbitrary vertices of $\Cay(G,S)$ in terms‎‎of complex irreducible representations of $G$‎. ‎Using our main result‎, ‎we give exact formulas for the number of walks‎‎of any length between two vertices in complete graphs‎, ‎cycles‎, ‎complete bipartite graphs‎, ‎Hamming graphs and complete transposition‎‎graphs‎.

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The main eigenvalues and number of walks in self-complementary graphs

Cited by 5 publications

References 7 publications

On strongly asymmetric and controllable primitive graphs

On strongly asymmetric and controllable primitive graphs

Balanced Centrality of Networks

On the Walks on Cayley Graphs‎

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