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The number of positive, negative and zero eigenvalues of the adjacency matrix of $\\Gamma$ are called positive inertia index, negative inertia index and nullity of $\\Gamma$, which are denoted by $i_+(\\Gamma)$, $i_-(\\Gamma)$ and $\\eta(\\Gamma)$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $\\Gamma$. We study relationships between the girth and the negative inertia index of $\\Gamma$ in this article. We prove $i_{-}(\\Gamma)\\geq \\lceil\\frac{g}{2}\\rceil-1$ and extremal signed"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2505.08539","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2025-05-13T13:10:19Z","cross_cats_sorted":[],"title_canon_sha256":"f273e47e5917924a666b122ec873675cd049b2559f7b7409fb41288f3b58d36d","abstract_canon_sha256":"c46c343cd5bfe2f8f52cae0fb97897e0c8a4acd048e218598c4519c1bb017748"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:02:32.032403Z","signature_b64":"SRXO2lg1E5tCkt2wpqqcvVWOYtIvKqPqEphPMcjc40GsEtcSOYhen7rXDB/XnjdLgykfpatUOg5CNrh43WDUAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"402547cce2ca5b65eefcb9e92864454f1a46519ff6127f8368675c875742ea3e","last_reissued_at":"2026-07-05T11:02:32.031901Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:02:32.031901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connected signed graphs with given inertia indices and given girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Beiyan Liu, Fang Duan","submitted_at":"2025-05-13T13:10:19Z","abstract_excerpt":"Suppose that $\\Gamma=(G, \\sigma)$ is a connected signed graph with at least one cycle. 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