ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P4.08
https://doi.org/10.26493/1855-3974.3180.7ea
(Also available at http://amc-journal.eu)
On the eigenvalues of complete bipartite signed
graphs*
Shariefuddin Pirzada † , Tahir Shamsher ‡
Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India
Mushtaq A. Bhat
Department of Mathematics, National Institute of Technology, Srinagar, India
Received 1 August 2023, accepted 30 January 2024, published online 9 October 2024
Abstract
Let Γ = (G, σ) be a signed graph, where σ is the sign function on the edges of G.
The adjacency matrix of Γ is defined canonically. Let (Kp,q, σ), p ≤ q, be a complete
bipartite signed graph with bipartition (Up, Vq), where Up = {u1, u2, . . . , up} and Vq =
{v1, v2, . . . , vq}. Let (Kp,q, σ)[Ur ∪ Vs], r ≤ p and s ≤ q, be an induced signed subgraph
on minimum vertices r+s, which contains all negative edges of the signed graph (Kp,q, σ).
In this paper, we show that the nullity of the signed graph (Kp,q, σ) is at least p+q−2k−2,
where k = min(r, s). The spectrum of a complete bipartite signed graph whose negative
edges induce either a disjoint complete bipartite subgraphs or a path is determined. Finally,
we obtain the spectrum of a complete bipartite signed graph whose negative edges (positive
edges) induce a regular subgraph H . It turns out that there is a relationship between the
eigenvalues of this complete bipartite signed graph and the non-negative eigenvalues of H .
Keywords: Signed graph, adjacency matrix, nullity, spectrum of complete bipartite graph.
Math. Subj. Class. (2020): 05C22, 05C50
*The authors are grateful to the anonymous referee for the valuable comments, which has considerably im-
proved the presentation of the paper.
†Corresponding author. The research of S. Pirzada is supported by SERB-DST research project number
CRG/2020/000109.
‡The research of Tahir Shamsher is supported by SRF financial assistance by Council of Scientific and Indus-
trial Research (CSIR), New Delhi, India.
E-mail addresses: pirzadasd@kashmiruniversity.ac.in (Shariefuddin Pirzada), tahir.maths.uok@gmail.com
(Tahir Shamsher), mushtaqab@nitsri.net (Mushtaq A. Bhat)
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 24 (2024) #P4.08
https://doi.org/10.26493/1855-3974.3180.7ea
(Dostopno tudi na http://amc-journal.eu)
Lastne vrednosti polnih dvodelnih predznačenih
grafov*
Shariefuddin Pirzada † , Tahir Shamsher ‡
Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India
Mushtaq A. Bhat
Department of Mathematics, National Institute of Technology, Srinagar, India
Prejeto 1. avgusta 2023, sprejeto 30. januarja 2024, objavljeno na spletu 9. oktobra 2024
Povzetek
Naj bo Γ = (G, σ) predznačeni graf, kjer je σ funkcija, ki vsaki povezavi grafa G
priredi njen predznak. Matrika sosednosti grafa Γ je definirana kanonično.
Naj bo (Kp,q, σ), p ≤ q polni dvodelni označeni graf z biparticijo (Up, Vq), kjer je
Up = {u1, u2, . . . , up} in Vq = {v1, v2, . . . , vq}. Naj bo (Kp,q, σ)[Ur ∪ Vs], kjer je
r ≤ p in s ≤ q, inducirani predznačeni podgraf na najmanj r + s točkah, ki vsebuje vse
negativne povezave predznačenega grafa (Kp,q, σ). V tem članku pokažemo, da je ničnost
predznačenega grafa (Kp,q, σ) najmanj p + q − 2k − 2, kjer je k = min(r, s). Določimo
spekter polnega dvodelnega predznačenega grafa, katerega negativne povezave inducirajo
bodisi disjunktne polne dvodelne podgrafe bodisi pot. Nazadnje, določimo spekter polnega
dvodelnega predznačenega grafa, katerega negativne povezave (ali pa pozitivne povezave)
inducirajo regularen podgraf H . Izkaže se, da obstaja zveza med lastnimi vrednostmi tega
polnega dvodelnega predznačenega grafa in nenegatvnimi lastnimi vrednostmi grafa H .
Ključne besede: Predznačeni graf, matrika sosednosti, ničnost, spekter polnega dvodelnega grafa.
Math. Subj. Class. (2020): 05C22, 05C50
*Avtorji se zahvaljujejo neznanemu recenzentu za dragocene pripombe, ki so znatno izboljšale razumljivost
članka.
†Kontaktni avtor. Raziskava S. Pirzada je podprta s strani SERB-DST, raziskovalni projekt št.
CRG/2020/000109.
‡Raziskavo Tahirja Shamsherja je s finančno pomočjo SRF podprl Svet za znanstvene in industrijske raziskave
(CSIR), New Delhi, Indija.
E-poštni naslovi: pirzadasd@kashmiruniversity.ac.in (Shariefuddin Pirzada), tahir.maths.uok@gmail.com
(Tahir Shamsher), mushtaqab@nitsri.net (Mushtaq A. Bhat)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/