ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P1.06
https://doi.org/10.26493/1855-3974.2697.43a
(Also available at http://amc-journal.eu)
On the Aα-spectral radius of connected graphs*
Abdollah Alhevaz , Maryam Baghipur
Faculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box: 316-3619995161, Shahrood, Iran
Hilal Ahmad Ganie
Department of School Education, JK Govt. Kashmir, India
Kinkar Chandra Das †
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Received 14 September 2021, accepted 23 June 2022, published online 5 October 2022
Abstract
For a simple graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G) =
αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the
diagonal matrix of the vertex degrees. It is clear that A0(G) = A(G) and 2A 1
2
(G) =
Q(G) implying that the matrix Aα(G) is a generalization of the adjacency matrix and the
signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for
the generalized adjacency spectral radius λ(Aα(G)), in terms of vertex degrees, average
vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are
characterized. We will show that our bounds are better than some of the already known
bounds for some classes of graphs. We derive a general upper bound for λ(Aα(G)), in
terms of vertex degrees and positive real numbers bi. As application, we obtain some new
upper bounds for λ(Aα(G)). Further, we obtain some relations between clique number
ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph
G.
Keywords: Adjacency matrix, signless Laplacian matrix, generalized adjacency matrix, spectral ra-
dius, degree sequence, clique number, independence number.
Math. Subj. Class. (2020): Primary: 05C50, 05C12; Secondary: 15A18.
*The authors would like to thank the handling editor and two anonymous referees for their detailed constructive
comments that helped improve the quality of the paper.
†Corresponding author. Partially supported by the National Research Foundation of the Korean government
with grant No. 2021R1F1A1050646.
E-mail addresses: a.alhevaz@shahroodut.ac.ir (Abdollah Alhevaz), maryamb8989@gmail.com (Maryam
Baghipur), hilahmad1119kt@gmail.com (Hilal Ahmad Ganie), kinkardas2003@gmail.com (Kinkar Chandra
Das)
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) #P1.06
https://doi.org/10.26493/1855-3974.2697.43a
(Dostopno tudi na http://amc-journal.eu)
Aα-spektralni polmer povezanih grafov*
Abdollah Alhevaz , Maryam Baghipur
Faculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box: 316-3619995161, Shahrood, Iran
Hilal Ahmad Ganie
Department of School Education, JK Govt. Kashmir, India
Kinkar Chandra Das †
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Prejeto 14. septembra 2021, sprejeto 23. junija 2022, objavljeno na spletu 5. oktobra 2022
Povzetek
Naj bo G enostaven graf; potem je posplošena matrika sosednosti Aα(G) definirana
kot Aα(G) = αD(G)+(1−α)A(G), α ∈ [0, 1], kjer je A(G) matrika sosednosti in D(G)
diagonalna matrika stopenj vozlišč. Jasno je, da A0(G) = A(G) in 2A 1
2
(G) = Q(G), kar
pomeni, da je matrika Aα(G) posplošitev matrike sosednosti in nepredznačene Laplaceove
matrike. V tem članku izpeljemo nekaj novih zgornjih in spodnjih mej za spektralni polmer
λ(Aα(G)) posplošene matrike sosednosti glede na stopnje vozlišč, povprečne 2-stopnje vo-
zlišč, red, velikost, itd. Karakteriziramo ekstremne grafe, ki te meje dosegajo. Pokažemo,
da so naše meje boljše od nekaterih že znanih mej za nekatere razrede grafov. Izpeljemo
splošno zgornjo mejo za λ(Aα(G)) glede na stopnje vozlišč in pozitivna realna števila bi.
Z uporabo teh rezultatov dobimo za λ(Aα(G)) nekaj novih zgornjih mej. Izpeljemo tudi
nekaj relacij med številom klik ω(G), številom neodvisnosti γ(G) in lastnimi vrednostmi
grafa G za pripadajočo posplošeno matriko sosednosti.
Ključne besede: Matrika sosednosti, nepredznačena Laplaceova matrika, posplošena matrika sosed-
nosti, spektralni polmer, zaporedje stopenj, število klik, število neodvisnosti.
Math. Subj. Class. (2020): Primarna: 05C50, 05C12; sekundarna: 15A18.
*Avtorji bi se radi zahvalili uredniku za njegovo delo in dvema recenzentoma za njune podrobne konstruktivne
komentarje, ki so pomagali izboljšati kakovost članka.
†Kontaktni avtor. Delno podprt s strani National Research Foundation of the Korean government z dotacijo št.
2021R1F1A1050646.
E-poštni naslovi: a.alhevaz@shahroodut.ac.ir (Abdollah Alhevaz), maryamb8989@gmail.com (Maryam
Baghipur), hilahmad1119kt@gmail.com (Hilal Ahmad Ganie), kinkardas2003@gmail.com (Kinkar Chandra
Das)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/