{"title":"计算图的拉普拉斯能量、类拉普拉斯能量不变量和Kirchhoff指数","authors":"S. Bhatnagar, Merajuddin, S. Pirzada","doi":"10.2478/ausi-2022-0011","DOIUrl":null,"url":null,"abstract":"Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as LE=∑i=1n| μi-2mn | LE = \\sum\\nolimits_{i = 1}^n {\\left| {{\\mu _i} - {{2m} \\over n}} \\right|} , LEL=∑i=1n-1μi LEL = \\sum\\nolimits_{i = 1}^{n - 1} {\\sqrt {{\\mu _i}} } and Kf=n∑i=1n-11μi Kf = n\\sum\\nolimits_{i = 1}^{n - 1} {{1 \\over {{\\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.","PeriodicalId":41480,"journal":{"name":"Acta Universitatis Sapientiae Informatica","volume":"49 1","pages":"185 - 198"},"PeriodicalIF":0.3000,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs\",\"authors\":\"S. Bhatnagar, Merajuddin, S. Pirzada\",\"doi\":\"10.2478/ausi-2022-0011\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as LE=∑i=1n| μi-2mn | LE = \\\\sum\\\\nolimits_{i = 1}^n {\\\\left| {{\\\\mu _i} - {{2m} \\\\over n}} \\\\right|} , LEL=∑i=1n-1μi LEL = \\\\sum\\\\nolimits_{i = 1}^{n - 1} {\\\\sqrt {{\\\\mu _i}} } and Kf=n∑i=1n-11μi Kf = n\\\\sum\\\\nolimits_{i = 1}^{n - 1} {{1 \\\\over {{\\\\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.\",\"PeriodicalId\":41480,\"journal\":{\"name\":\"Acta Universitatis Sapientiae Informatica\",\"volume\":\"49 1\",\"pages\":\"185 - 198\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Universitatis Sapientiae Informatica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/ausi-2022-0011\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Universitatis Sapientiae Informatica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/ausi-2022-0011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Computing Laplacian energy, Laplacian-energy-like invariant and Kirchhoff index of graphs
Abstract Let G be a simple connected graph of order n and size m. The matrix L(G)= D(G)− A(G) is called the Laplacian matrix of the graph G,where D(G) and A(G) are the degree diagonal matrix and the adjacency matrix, respectively. Let the vertex degree sequence be d1 ≥ d2 ≥··· ≥ dn and let μ1 ≥ μ2 ≥··· ≥ μn−1 >μn = 0 be the eigenvalues of the Laplacian matrix of G. The graph invariants, Laplacian energy (LE), the Laplacian-energy-like invariant (LEL) and the Kirchhoff index (Kf), are defined in terms of the Laplacian eigenvalues of graph G, as LE=∑i=1n| μi-2mn | LE = \sum\nolimits_{i = 1}^n {\left| {{\mu _i} - {{2m} \over n}} \right|} , LEL=∑i=1n-1μi LEL = \sum\nolimits_{i = 1}^{n - 1} {\sqrt {{\mu _i}} } and Kf=n∑i=1n-11μi Kf = n\sum\nolimits_{i = 1}^{n - 1} {{1 \over {{\mu _i}}}} respectively. In this paper, we obtain a new bound for the Laplacian-energy-like invariant LEL and establish the relations between Laplacian-energy-like invariant LEL and the Kirchhoff index Kf.Further,weobtain the relations between the Laplacian energy LE and Kirchhoff index Kf.