M. Arockiaraj, Shagufa Mushtaq, S. Klavžar, J. C. Fiona, K. Balasubramanian
The Szeged index is a bond-additive topological descriptor that quantifies each bond’s terminal atoms based on their closeness sets which is measured by multiplying the number of atoms in the closeness sets. Based on the high correlation between the Szeged index and physico-chemical properties of chemical compounds, Szeged-like indices have been proposed by considering closeness sets with bond counts and other mathematical operations like addition and subtraction. As there are many ways to compute the Szeged-like indices, the cut method is predominantly used due to its complexity compared to other approaches based on algorithms and interpolations. Yet, we here analyze the usefulness of the cut method in the case of melem structures and find that it is less effective when the size and shape of the cavities change in the structures.
{"title":"Szeged-Like Topological Indices and the Efficacy of the Cut Method: The Case of Melem Structures","authors":"M. Arockiaraj, Shagufa Mushtaq, S. Klavžar, J. C. Fiona, K. Balasubramanian","doi":"10.47443/dml.2021.s209","DOIUrl":"https://doi.org/10.47443/dml.2021.s209","url":null,"abstract":"The Szeged index is a bond-additive topological descriptor that quantifies each bond’s terminal atoms based on their closeness sets which is measured by multiplying the number of atoms in the closeness sets. Based on the high correlation between the Szeged index and physico-chemical properties of chemical compounds, Szeged-like indices have been proposed by considering closeness sets with bond counts and other mathematical operations like addition and subtraction. As there are many ways to compute the Szeged-like indices, the cut method is predominantly used due to its complexity compared to other approaches based on algorithms and interpolations. Yet, we here analyze the usefulness of the cut method in the case of melem structures and find that it is less effective when the size and shape of the cavities change in the structures.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48396938","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}
The Wiener index W (G) of a graph G is the sum of distances between all vertices of G. The Wiener index of a family of connected graphs is defined as the sum of the Wiener indices of its members. Two families of graphs can be constructed by identifying a fixed vertex of an arbitrary graph F with vertices or subdivision vertices of an arbitrary tree T of order n. Let Gv be a graph obtained by identifying a fixed vertex of F with a vertex v of T . The first family V = {Gv | v ∈ V (T )} contains n graphs. Denote by Gve a graph obtained by identifying the same fixed vertex of F with the subdivision vertex ve of an edge e in T . The second family E = {Gve | e ∈ E(T )} contains n − 1 graphs. It is proved that W (V) = W (E) if and only if W (F ) = 2W (T ).
{"title":"On the Wiener Index of Two Families Generated by Joining a Graph to a Tree","authors":"A. Dobrynin","doi":"10.47443/dml.2021.s208","DOIUrl":"https://doi.org/10.47443/dml.2021.s208","url":null,"abstract":"The Wiener index W (G) of a graph G is the sum of distances between all vertices of G. The Wiener index of a family of connected graphs is defined as the sum of the Wiener indices of its members. Two families of graphs can be constructed by identifying a fixed vertex of an arbitrary graph F with vertices or subdivision vertices of an arbitrary tree T of order n. Let Gv be a graph obtained by identifying a fixed vertex of F with a vertex v of T . The first family V = {Gv | v ∈ V (T )} contains n graphs. Denote by Gve a graph obtained by identifying the same fixed vertex of F with the subdivision vertex ve of an edge e in T . The second family E = {Gve | e ∈ E(T )} contains n − 1 graphs. It is proved that W (V) = W (E) if and only if W (F ) = 2W (T ).","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45119899","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}
Abstract In this paper, the author considers the twisted q-analogues of Catalan numbers, which are arisen from the fermionic p-adic q-integrals. By using the fermionic p-adic q-integrals or generating functions, some explicit identities and properties for the twisted q-analogues of Catalan numbers and polynomials are given.
{"title":"Some Explicit Expressions for Twisted q-Analogues of Catalan Numbers and Polynomials","authors":"D. Lim","doi":"10.47443/dml.2022.0007","DOIUrl":"https://doi.org/10.47443/dml.2022.0007","url":null,"abstract":"Abstract In this paper, the author considers the twisted q-analogues of Catalan numbers, which are arisen from the fermionic p-adic q-integrals. By using the fermionic p-adic q-integrals or generating functions, some explicit identities and properties for the twisted q-analogues of Catalan numbers and polynomials are given.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45803141","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}
Abstract For a graph G, the geometric-arithmetic index of G, denoted by GA(G), is defined as the sum of the quantities 2 √ dx × dy/(dx + dy) over all edges xy ∈ E(G). Here, dx indicates the vertex degree of x. For every tree T of order n ≥ 3, Vukičević and Furtula [J. Math. Chem. 46 (2009) 1369–1376] demonstrated that GA(T ) ≤ 4 √ 2 3 + (n − 3). This result is extended in the present paper. Particularly, for any tree T of order n ≥ 5 and maximum degree ∆, it is proved that
{"title":"Extremal Trees for the Geometric-Arithmetic Index with the Maximum Degree","authors":"A. Divya, A. Manimaran","doi":"10.47443/dml.2021.s207","DOIUrl":"https://doi.org/10.47443/dml.2021.s207","url":null,"abstract":"Abstract For a graph G, the geometric-arithmetic index of G, denoted by GA(G), is defined as the sum of the quantities 2 √ dx × dy/(dx + dy) over all edges xy ∈ E(G). Here, dx indicates the vertex degree of x. For every tree T of order n ≥ 3, Vukičević and Furtula [J. Math. Chem. 46 (2009) 1369–1376] demonstrated that GA(T ) ≤ 4 √ 2 3 + (n − 3). This result is extended in the present paper. Particularly, for any tree T of order n ≥ 5 and maximum degree ∆, it is proved that","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48493245","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}
S. Altindag, I. Milovanovic, E. Milovanovic, M. Matejic
Abstract For a connected graphGwithn vertices andm edges, the degree Kirchhoff index ofG is defined asKf∗ (G) = 2m ∑n−1 i=1 (γi) , where γ1 ≥ γ2 ≥ · · · ≥ γn−1 > γn = 0 are the normalized Laplacian eigenvalues of G. In this paper, a lower bound on the degree Kirchhoff index of bipartite graphs is established. Also, it is proved that the obtained bound is stronger than a lower bound derived by Zhou and Trinajstić in [J. Math. Chem. 46 (2009) 283–289].
{"title":"An Improved Lower Bound for the Degree Kirchhoff Index of Bipartite Graphs","authors":"S. Altindag, I. Milovanovic, E. Milovanovic, M. Matejic","doi":"10.47443/dml.2021.0118","DOIUrl":"https://doi.org/10.47443/dml.2021.0118","url":null,"abstract":"Abstract For a connected graphGwithn vertices andm edges, the degree Kirchhoff index ofG is defined asKf∗ (G) = 2m ∑n−1 i=1 (γi) , where γ1 ≥ γ2 ≥ · · · ≥ γn−1 > γn = 0 are the normalized Laplacian eigenvalues of G. In this paper, a lower bound on the degree Kirchhoff index of bipartite graphs is established. Also, it is proved that the obtained bound is stronger than a lower bound derived by Zhou and Trinajstić in [J. Math. Chem. 46 (2009) 283–289].","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46653300","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}
Skew Dyck paths are like Dyck paths, but an additional south-west step (−1,−1) is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used.
{"title":"Skew Dyck Paths With Catastrophes","authors":"H. Prodinger","doi":"10.47443/dml.2022.008","DOIUrl":"https://doi.org/10.47443/dml.2022.008","url":null,"abstract":"Skew Dyck paths are like Dyck paths, but an additional south-west step (−1,−1) is allowed, provided that the path does not intersect itself. Lattice paths with catastrophes can drop from any level to the origin in just one step. We combine these two ideas. The analysis is strictly based on generating functions, and the kernel method is used.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45895841","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}
A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, . . . , n} to the vertices of G. The strength of G is defined by str (G) = min {strf (G) |f is a numbering of G} , where strf (G) = max {f (u) + f (v) |uv ∈ E (G)}. In this paper, we present some results obtained from factorizations of complete graphs. In particular, we show that for every k ∈ [1, n− 1], there exists a graph G of order n satisfying δ (G) = k and str (G) = n+ k, where δ (G) denotes the minimum degree of G.
{"title":"A Result on the Strength of Graphs by Factorizations of Complete Graphs","authors":"Rikio Ichishima, F. Muntaner-Batle","doi":"10.47443/dml.2021.0096","DOIUrl":"https://doi.org/10.47443/dml.2021.0096","url":null,"abstract":"A numbering f of a graph G of order n is a labeling that assigns distinct elements of the set {1, 2, . . . , n} to the vertices of G. The strength of G is defined by str (G) = min {strf (G) |f is a numbering of G} , where strf (G) = max {f (u) + f (v) |uv ∈ E (G)}. In this paper, we present some results obtained from factorizations of complete graphs. In particular, we show that for every k ∈ [1, n− 1], there exists a graph G of order n satisfying δ (G) = k and str (G) = n+ k, where δ (G) denotes the minimum degree of G.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41619877","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}
{"title":"When the Cartesian product of directed cycles is hyper-Hamiltonian","authors":"","doi":"10.47443/dml.2021.0088","DOIUrl":"https://doi.org/10.47443/dml.2021.0088","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49049641","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}
For a graph G, and for two distinct vertices u and v of G, let nG(u, v) be the number of vertices of G that are closer in G to u than to v. The distance-unbalancedness of G is the sum of |nG(u, v)− nG(v, u)| over all unordered pairs of distinct vertices u and v of G. We determine the minimum distance-unbalancedness of 2-self-centered graphs with given number of edges. We also determine the minimum distance-unbalancedness of graphs with at least one universal vertex and given number of edges.
{"title":"Minimum Distance-Unbalancedness of Graphs With Diameter 2 and Given Number of Edges","authors":"Kexiang Xu, Peiqi Yao","doi":"10.47443/dml.2021.s205","DOIUrl":"https://doi.org/10.47443/dml.2021.s205","url":null,"abstract":"For a graph G, and for two distinct vertices u and v of G, let nG(u, v) be the number of vertices of G that are closer in G to u than to v. The distance-unbalancedness of G is the sum of |nG(u, v)− nG(v, u)| over all unordered pairs of distinct vertices u and v of G. We determine the minimum distance-unbalancedness of 2-self-centered graphs with given number of edges. We also determine the minimum distance-unbalancedness of graphs with at least one universal vertex and given number of edges.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42515027","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}
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