<p>This paper presents new results on the edge irregularity strength of various graph classes. The exact values are determined for the series composition of the isomorphic uniform theta graphs, expressed as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta (n;m;r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>m</mi> <mo>;</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, in particular for the cases where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n = m = 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n = 4, m = 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, the edge irregularity strength is obtained for the shadow graph of paths with odd length; and the square of the path <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$P_6$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation> and the theta graph <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\theta (1,4,4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, each extended by adding a single pendant edge. Furthermore, the paper explores an application of edge irregularity strength in optimizing path selection within communication networks.</p>

Edge Irregularity Strength of Graphs: Theoretical Results and Applications to Communication Networks

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发表日期：

2026年7月10日

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Umme Salma,
H. M. Nagesh,
U. Vijaya Chandra Kumar,
N. Narahari

摘要

This paper presents new results on the edge irregularity strength of various graph classes. The exact values are determined for the series composition of the isomorphic uniform theta graphs, expressed as $\theta (n;m;r)$ $θ (n; m; r)$ , in particular for the cases where $n = m = 3$ $n = m = 3$ and $n = 4, m = 3$ $n = 4, m = 3$ with $r \ge 2$ $r \geq 2$ . Additionally, the edge irregularity strength is obtained for the shadow graph of paths with odd length; and the square of the path $P_6$ $P_{6}$ and the theta graph $\theta (1,4,4)$ $θ (1, 4, 4)$ , each extended by adding a single pendant edge. Furthermore, the paper explores an application of edge irregularity strength in optimizing path selection within communication networks.