Line and Subdivision Graphs Determined by <inline-formula> <mml:math id="mm999" display="block"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="double-struck">T</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:semantics> </mml:math> </inline-formula>-Gain Graphs
oleh: Abdullah Alazemi, Milica Anđelić, Francesco Belardo, Maurizio Brunetti, Carlos M. da Fonseca
| Format: | Article |
|---|---|
| Diterbitkan: | MDPI AG 2019-10-01 |
Deskripsi
Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="double-struck">T</mi> <mn>4</mn> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>i</mi> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be the subgroup of fourth roots of unity inside <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">T</mi> </semantics> </math> </inline-formula>, the multiplicative group of complex units. For a <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">T</mi> <mn>4</mn> </msub> </semantics> </math> </inline-formula>-gain graph <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="sans-serif">Φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Γ</mi> <mo>,</mo> <msub> <mi mathvariant="double-struck">T</mi> <mn>4</mn> </msub> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>, we introduce gain functions on its line graph <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Γ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> and on its subdivision graph <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Γ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>. The corresponding gain graphs <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Φ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Φ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph <inline-formula> <math display="inline"> <semantics> <mi mathvariant="sans-serif">Φ</mi> </semantics> </math> </inline-formula>, and the adjacency characteristic polynomials of <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Φ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">S</mi> <mo stretchy="false">(</mo> <mi mathvariant="sans-serif">Φ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula>. A suitably defined incidence matrix for <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">T</mi> <mn>4</mn> </msub> </semantics> </math> </inline-formula>-gain graphs plays an important role in this context.