| Summary: | In this paper, we introduce a new graph operation called <i>subdivision vertex-edge join</i> (denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>G</mi> <mn>1</mn> <mi>S</mi> </msubsup> <mo>▹</mo> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mn>2</mn> <mi>V</mi> </msubsup> <mo>∪</mo> <msubsup> <mi>G</mi> <mn>3</mn> <mi>E</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for short), and then the <i>adjacency spectrum</i>, the <i>Laplacian spectrum</i> and the <i>signless Laplacian spectrum</i> of <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>G</mi> <mn>1</mn> <mi>S</mi> </msubsup> <mo>▹</mo> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mn>2</mn> <mi>V</mi> </msubsup> <mo>∪</mo> <msubsup> <mi>G</mi> <mn>3</mn> <mi>E</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are respectively determined in terms of the corresponding spectra for a regular graph <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> and two arbitrary graphs <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mn>3</mn> </msub> </semantics> </math> </inline-formula>. All the above can be viewed as the generalizations of the main results in [X. Liu, Z. Zhang, <i>Bull. Malays. Math. Sci. Soc.</i>, 2017:1⁻17]. Furthermore, we also determine the <i>normalized Laplacian spectrum</i> of <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>G</mi> <mn>1</mn> <mi>S</mi> </msubsup> <mo>▹</mo> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mn>2</mn> <mi>V</mi> </msubsup> <mo>∪</mo> <msubsup> <mi>G</mi> <mn>3</mn> <mi>E</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> whenever <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics> </math> </inline-formula> are regular graphs for each index <inline-formula> <math display="inline"> <semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>. As applications, we construct infinitely many pairs of <i>A-cospectral mates</i>, <i>L-cospectral mates</i>, <i>Q-cospectral mates</i> and <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">L</mi> </semantics> </math> </inline-formula>-<i>cospectral mates</i>. Finally, we give the number of <i>spanning trees</i>, the (<i>degree-</i>)<i>Kirchhoff index</i> and the <i>Kemeny’s constant</i> of <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>G</mi> <mn>1</mn> <mi>S</mi> </msubsup> <mo>▹</mo> <mrow> <mo>(</mo> <msubsup> <mi>G</mi> <mn>2</mn> <mi>V</mi> </msubsup> <mo>∪</mo> <msubsup> <mi>G</mi> <mn>3</mn> <mi>E</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, respectively.
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