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The minimum exponential atom-bond connectivity energy of trees 树的最小指数原子键连接能
IF 0.5
Special Matrices Pub Date : 2024-01-01 DOI: 10.1515/spma-2023-0108
Wei Gao
{"title":"The minimum exponential atom-bond connectivity energy of trees","authors":"Wei Gao","doi":"10.1515/spma-2023-0108","DOIUrl":"https://doi.org/10.1515/spma-2023-0108","url":null,"abstract":"Abstract Let G = ( V ( G ) , E ( G ) ) G=left(Vleft(G),Eleft(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) of G G is an n × n ntimes n matrix whose ( i , j ) left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{sqrt{tfrac{dleft({v}_{i})+dleft({v}_{j})-2}{dleft({v}_{i})dleft({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}in Eleft(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{rm{ABC}}}}left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"31 7","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139455025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Refined inertias of positive and hollow positive patterns 积极和空洞积极模式的精炼惰性
IF 0.5
Special Matrices Pub Date : 2024-01-01 DOI: 10.1515/spma-2023-0107
Adam H. Berliner, M. Catral, D. Olesky, P. van den Driessche
{"title":"Refined inertias of positive and hollow positive patterns","authors":"Adam H. Berliner, M. Catral, D. Olesky, P. van den Driessche","doi":"10.1515/spma-2023-0107","DOIUrl":"https://doi.org/10.1515/spma-2023-0107","url":null,"abstract":"Abstract We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}ge 1 and n − ≥ 2 {n}_{-}ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ nto infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139393406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The perturbation of Drazin inverse and dual Drazin inverse Drazin 逆和双重 Drazin 逆的扰动
IF 0.5
Special Matrices Pub Date : 2024-01-01 DOI: 10.1515/spma-2023-0110
Hongxing Wang, Chong Cui, Yimin Wei
{"title":"The perturbation of Drazin inverse and dual Drazin inverse","authors":"Hongxing Wang, Chong Cui, Yimin Wei","doi":"10.1515/spma-2023-0110","DOIUrl":"https://doi.org/10.1515/spma-2023-0110","url":null,"abstract":"Abstract In this study, we derive the Drazin inverse ( A + ε B ) D {left(A+varepsilon B)}^{D} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) > 1 {rm{Ind}}left(A+varepsilon B)gt 1 and Ind ( A ) = k {rm{Ind}}left(A)=k and the group inverse ( A + ε B ) # {left(A+varepsilon B)}^{#} of the complex matrix A + ε B A+varepsilon B with Ind ( A + ε B ) = 1 {rm{Ind}}left(A+varepsilon B)=1 and Ind ( A ) = k {rm{Ind}}left(A)=k when ε B varepsilon B is viewed as the perturbation of A A . If the dual Drazin inverse (DDGI) A ^ DDGI {widehat{A}}^{{rm{DDGI}}} of A ^ widehat{A} is considered as a notation. We calculate ( A + ε B ) D − A ^ DDGI {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}} and ( A + ε B ) # − A ^ DDGI {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}} and obtain ‖ ( A + ε B ) D − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{D}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) and ‖ ( A + ε B ) # − A ^ DDGI ‖ P ∈ O ( ε 2 ) Vert {left(A+varepsilon B)}^{#}-{widehat{A}}^{{rm{DDGI}}}{Vert }_{P}in {mathcal{O}}left({varepsilon }^{2}) . Meanwhile, we give some examples to verify these conclusions.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"11 12","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139457459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On inverse sum indeg energy of graphs 图的逆和指数能
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2022-0175
Fareeha Jamal, Muhammad Imran, B. Rather
{"title":"On inverse sum indeg energy of graphs","authors":"Fareeha Jamal, Muhammad Imran, B. Rather","doi":"10.1515/spma-2022-0175","DOIUrl":"https://doi.org/10.1515/spma-2022-0175","url":null,"abstract":"Abstract For a simple graph with vertex set { v 1 , v 2 , … , v n } left{{v}_{1},{v}_{2},ldots ,{v}_{n}right} and degree sequence d v i i = 1 , 2 , … , n {d}_{{v}_{i}}hspace{0.33em}i=1,2,ldots ,n , the inverse sum indeg matrix (ISI matrix) A ISI ( G ) = ( a i j ) {A}_{{rm{ISI}}}left(G)=left({a}_{ij}) of G G is a square matrix of order n , n, where a i j = d v i d v j d v i + d v j , {a}_{ij}=frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if v i {v}_{i} is adjacent to v j {v}_{j} and 0, otherwise. The multiset of eigenvalues τ 1 ≥ τ 2 ≥ ⋯ ≥ τ n {tau }_{1}ge {tau }_{2}hspace{0.33em}ge cdots ge {tau }_{n} of A ISI ( G ) {A}_{{rm{ISI}}}left(G) is known as the ISI spectrum of G G . The ISI energy of G G is the sum ∑ i = 1 n ∣ τ i ∣ mathop{sum }limits_{i=1}^{n}| {tau }_{i}| of the absolute ISI eigenvalues of G . G. In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n ≥ 9 nge 9 .","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47014970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the distance energy of k-uniform hypergraphs 关于k-均匀超图的距离能
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2023-0188
Kshitij Sharma, S. Panda
{"title":"On the distance energy of k-uniform hypergraphs","authors":"Kshitij Sharma, S. Panda","doi":"10.1515/spma-2023-0188","DOIUrl":"https://doi.org/10.1515/spma-2023-0188","url":null,"abstract":"Abstract In this article, we extend the concept of distance energy for hypergraphs. We first establish a relation between the distance energy and the distance spectral radius. Then, we obtain some bounds for the distance energy in terms of some invariant of hypergraphs such as the determinant of the distance matrix, number of vertices, and Wiener index along with the distance energy of join of k k -uniform hypergraphs. Furthermore, it is shown that the determinant of the distance matrix of k k -uniform hyperstar on n n vertices is ( − 1 ) n − 1 ( n − 1 ) k n − k k − 1 {left(-1)}^{n-1}left(n-1){k}^{tfrac{n-k}{k-1}} . Later, the distance spectrum of k k -uniform hyperstar is obtained, which gives the explicit distance energy of k k -uniform hyperstar.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45813795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Corrigendum to “Spectra universally realizable by doubly stochastic matrices” “双随机矩阵普遍可实现的谱”的勘误
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2022-0179
M. Collao, Mario Salas, R. Soto
{"title":"Corrigendum to “Spectra universally realizable by doubly stochastic matrices”","authors":"M. Collao, Mario Salas, R. Soto","doi":"10.1515/spma-2022-0179","DOIUrl":"https://doi.org/10.1515/spma-2022-0179","url":null,"abstract":"Abstract We correct an error in the statement and the proof of Theorem 2.1 and Corollary 2.1 in our previous study [Spec. Matrices 6 (2018), 301–309], Section 2: On nonnegative matrices similar to positive matrices.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47480725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On 3-by-3 row stochastic matrices 在3 × 3行随机矩阵上
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2023-0103
Nhi Pham, Ilya M. Spitkovsky
{"title":"On 3-by-3 row stochastic matrices","authors":"Nhi Pham, Ilya M. Spitkovsky","doi":"10.1515/spma-2023-0103","DOIUrl":"https://doi.org/10.1515/spma-2023-0103","url":null,"abstract":"Abstract The known constructive tests for the shapes of the numerical ranges in the 3-by-3 case are further specified when the matrices in question are row stochastic. Auxiliary results on the unitary (ir)reducibility of such matrices are also obtained.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Class of finite-dimensional matrices with diagonals that majorize their spectrum 一类有限维矩阵，其对角线使其谱最大化
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2022-0185
Jeffrey Uhlmann
{"title":"Class of finite-dimensional matrices with diagonals that majorize their spectrum","authors":"Jeffrey Uhlmann","doi":"10.1515/spma-2022-0185","DOIUrl":"https://doi.org/10.1515/spma-2022-0185","url":null,"abstract":"Abstract We define a special class of finite-dimensional matrices for which the diagonal majorizes the spectrum. This is the first class of matrices known to have this property, although the reverse majorization (i.e., the spectrum majorizing the diagonal) was previously known to hold for unitarily diagonalizable (i.e., normal) matrices. Currently, these are the only known matrix classes that structurally provide a majorization relationship between their spectrum and diagonal.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46317526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Idempotent operator and its applications in Schur complements on Hilbert C*-module Hilbert C*模上的Schur补上的幂等算子及其应用
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2022-0187
M. M. Karizaki, Z. N. Moghani
{"title":"Idempotent operator and its applications in Schur complements on Hilbert C*-module","authors":"M. M. Karizaki, Z. N. Moghani","doi":"10.1515/spma-2022-0187","DOIUrl":"https://doi.org/10.1515/spma-2022-0187","url":null,"abstract":"Abstract The present study proves that T T is an idempotent operator if and only if R ( I − T ∗ ) ⊕ R ( T ) = X {mathcal{ {mathcal R} }}left(I-{T}^{ast })oplus {mathcal{ {mathcal R} }}left(T)={mathcal{X}} and ( T ∗ T ) † = ( T † ) 2 T {left({T}^{ast }T)}^{dagger }={left({T}^{dagger })}^{2}T . Based on the equivalent conditions of an idempotent operator and related results, it is possible to obtain an explicit formula for the Moore-Penrose inverse of 2-by-2 block idempotent operator matrix. For the 2-by-2 block operator matrix, Schur complements and generalized Schur complement are well known and studied. The range inclusions of operators and idempotency of operators are used to obtain new conditions under which we can compute the Moore-Penrose inverse of Schur complements and generalized Schur complements of operators.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48091159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Diagonal dominance and invertibility of matrices 矩阵的对角优势与可逆性
IF 0.5
Special Matrices Pub Date : 2023-01-01 DOI: 10.1515/spma-2022-0181
C. Johnson, C. Marijuán, M. Pisonero
{"title":"Diagonal dominance and invertibility of matrices","authors":"C. Johnson, C. Marijuán, M. Pisonero","doi":"10.1515/spma-2022-0181","DOIUrl":"https://doi.org/10.1515/spma-2022-0181","url":null,"abstract":"Abstract A weakly diagonally dominant matrix may or may not be invertible. We characterize, in terms of combinatorial structure and sign pattern when such a matrix is invertible, which is the common case. Examples are given.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45593451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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