| Summary: | Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> the diagonal matrix of the degrees of a graph <i>G</i>, respectively. For <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, the <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-matrix is the general adjacency and signless Laplacian spectral matrix having the form of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>α</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>D</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <mi>A</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. Clearly, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is the adjacent matrix and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2</mn> <msub> <mi>A</mi> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msub> </mrow> </semantics> </math> </inline-formula> is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mi>α</mi> </msub> </semantics> </math> </inline-formula>-spectral radius of a cactus graph with <i>n</i> vertices and <i>k</i> cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.
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