ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.02
https://doi.org/10.26493/1855-3974.2842.6b5
(Also available at http://amc-journal.eu)
Total graph of a signed graph
Francesco Belardo
Department of Mathematics and Applications, University of Naples Federico II,
I-80126 Naples, Italy
Zoran Stanić *
Faculty of Mathematics, University of Belgrade, Studentski trg 16,
11 000 Belgrade, Serbia
Thomas Zaslavsky
Department of Mathematical Sciences, Binghamton University, Binghamton, NY
13902-6000, United States
Received 10 March 2022, accepted 24 March 2022, published online 8 September 2022
Abstract
The total graph is built by joining the graph to its line graph by means of the incidences.
We introduce a similar construction for signed graphs. Under two similar definitions of the
line signed graph, we define the corresponding total signed graph and we show that it is
stable under switching. We consider balance, the frustration index and frustration number,
and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency
matrix of the total graph and the spectra of certain compositions, and we determine some
with exactly two main eigenvalues.
Keywords: Bidirected graph, signed line graph, signed total graph, graph eigenvalues, regular signed
graph, Cartesian product graph.
Math. Subj. Class. (2020): 05C22, 05C50, 05C76.
*Corresponding author. The research is partially supported by the Science Fund of the Republic of Serbia;
grant number 7749676: Spectrally Constrained Signed Graphs with Applications in Coding Theory and Control
Theory – SCSG-ctct.
E-mail addresses: fbelardo@unina.it (Francesco Belardo), zstanic@matf.bg.ac.rs (Zoran Stanić),
zaslav@math.binghamton.edu (Thomas Zaslavsky)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.02
https://doi.org/10.26493/1855-3974.2842.6b5
(Dostopno tudi na http://amc-journal.eu)
Združeni graf predznačenega grafa
Francesco Belardo
Department of Mathematics and Applications, University of Naples Federico II,
I-80126 Naples, Italy
Zoran Stanić *
Faculty of Mathematics, University of Belgrade, Studentski trg 16,
11 000 Belgrade, Serbia
Thomas Zaslavsky
Department of Mathematical Sciences, Binghamton University, Binghamton, NY
13902-6000, United States
Prejeto 10. marca 2022, sprejeto 24. marca 2022, objavljeno na spletu 8. septembra 2022
Povzetek
Združeni graf je zgrajen tako, da se graf združi z njegovim povezavnim grafom s po-
močjo incidenc. Podobno konstrukcijo vpeljemo za predznačene grafe. Z dvema podob-
nima definicijama povezavnega predznačenega grafa definiramo ustrezen združeni pred-
značeni graf in pokažemo, da se le-ta pri operaciji preklapljanja ohranja. Upoštevamo
ravnotežje, frustracijski indeks in frustracijsko število ter največjo lastno vrednost. V
regularnem primeru izračunamo spekter matrike sosednosti združenega grafa in spektre
določenih sestavov, in prikažemo tiste z natanko dvema glavnima lastnima vrednostima.
Ključne besede: Dvosmerni graf, predznačeni graf povezav, predznačeni združeni graf, lastne vred-
nosti grafa, pravilni predznačeni graf, graf kartezičnega produkta.
Math. Subj. Class. (2020): 05C22, 05C50, 05C76.
*Kontaktni avtor. Raziskava je delno podprta s strani Science Fund of the Republic of Serbia; številka dotacije
7749676: Spektralno omejeni predznačeni grafi z aplikacijami v teoriji kodiranja in teoriji nadzora – SCSG-ctct.
E-poštni naslovi: fbelardo@unina.it (Francesco Belardo), zstanic@matf.bg.ac.rs (Zoran Stanić),
zaslav@math.binghamton.edu (Thomas Zaslavsky)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/