Some results on $σ_{t}$-irregularity
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2024
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| author | Filipovski, Slobodan Dimitrov, Darko Knor, Martin Škrekovski, Riste |
| author_facet | Filipovski, Slobodan Dimitrov, Darko Knor, Martin Škrekovski, Riste |
| contents | The $σ_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $σ_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\' {e}ti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanović [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that $σ_{t}(G)=n^{2}\cdot Var(G)$, where $Var(G)$ is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum $σ_{t}$-irregularity. We show that among all connected graphs on $n$ vertices, the split graphs $S_{\lceil\frac{n}{4}\rceil, \lfloor\frac{3n}{4}\rfloor }$ and $S_{\lfloor\frac{n}{4}\rfloor, \lceil\frac{3n}{4}\rceil }$ have the maximum $σ_{t}$-irregularity, and among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor\frac{n}{4}(2-\sqrt{2})\rfloor, \lceil\frac{n}{4}(2+\sqrt{2})\rceil }$ or $K_{\lceil\frac{n}{4}(2-\sqrt{2})\rceil, \lfloor\frac{n}{4}(2+\sqrt{2})\rfloor }$ has the maximum sigma total index. Moreover, various upper and lower bounds for $σ_{t}$-irregularity are provided; in this direction we give a relation between the graph energy $\mathcal{E}(G)$ and sigma total index $σ_{t}(G)$ and give another proof of two results by Dimitrov and Stevanović. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $σ_{t}$ and $σ$. We conclude the paper with two conjectures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04881 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some results on $σ_{t}$-irregularity Filipovski, Slobodan Dimitrov, Darko Knor, Martin Škrekovski, Riste Combinatorics 05C35, 05C05, 05C09 The $σ_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $σ_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\' {e}ti [Appl. Math. Comput. 344-345 (2019) 107-115], and recently rediscovered by Dimitrov and Stevanović [Appl. Math. Comput. 441 (2023) 127709]. In this paper we remark that $σ_{t}(G)=n^{2}\cdot Var(G)$, where $Var(G)$ is the degree variance of the graph. Based on this observation, we characterize irregular graphs with maximum $σ_{t}$-irregularity. We show that among all connected graphs on $n$ vertices, the split graphs $S_{\lceil\frac{n}{4}\rceil, \lfloor\frac{3n}{4}\rfloor }$ and $S_{\lfloor\frac{n}{4}\rfloor, \lceil\frac{3n}{4}\rceil }$ have the maximum $σ_{t}$-irregularity, and among all complete bipartite graphs on $n$ vertices, either the complete bipartite graph $K_{\lfloor\frac{n}{4}(2-\sqrt{2})\rfloor, \lceil\frac{n}{4}(2+\sqrt{2})\rceil }$ or $K_{\lceil\frac{n}{4}(2-\sqrt{2})\rceil, \lfloor\frac{n}{4}(2+\sqrt{2})\rfloor }$ has the maximum sigma total index. Moreover, various upper and lower bounds for $σ_{t}$-irregularity are provided; in this direction we give a relation between the graph energy $\mathcal{E}(G)$ and sigma total index $σ_{t}(G)$ and give another proof of two results by Dimitrov and Stevanović. Applying Fiedler's characterization of the largest and the second smallest Laplacian eigenvalue of the graph, we also establish new relationships between $σ_{t}$ and $σ$. We conclude the paper with two conjectures. |
| title | Some results on $σ_{t}$-irregularity |
| topic | Combinatorics 05C35, 05C05, 05C09 |
| url | https://arxiv.org/abs/2411.04881 |